On optimal k-fold colorings of webs and antiwebs

On optimal k-fold colorings of webs and antiwebs

Manoel Campˆelo^a,1,∗, Ricardo C. Corrˆea^b, Phablo F. S. Moura^c,1,2, Marcio C. Santos^b,3

aUniversidade Federal do Cear´a, Departamento de Estat´ıstica e Matem´atica Aplicada, Campus do Pici, Bloco 910, 60440-554 Fortaleza - CE, Brazil

bUniversidade Federal do Cear´a, Departamento de Computa¸c˜ao, Campus do Pici, Bloco 910, 60440-554 Fortaleza - CE, Brazil

cUniversidade de S˜ao Paulo, Instituto de Matem´atica e Estat´ıstica, Rua do Mat˜ao 1010, 05508-090 S˜ao Paulo - SP, Brazil

Abstract

A k-fold x-coloring of a graph is an assignment of (at least)k distinct colors from the set{1,2, . . . , x}to each vertex such that any two adjacent vertices are assigned disjoint sets of colors. The smallest numberxsuch thatGadmits ak-fold x-coloring is thek-th chromatic number of G, denoted by χk(G). We determine the exact value of this pa- rameter whenGis a web or an antiweb. Our results generalize the known corresponding results for odd cycles and imply necessary and sufficient conditions under whichχk(G) attains its lower and upper bounds based on clique and integer and fractional chromatic numbers. Additionally, we extend the concept ofχ-critical graphs toχ_k-critical graphs.

We identify the webs and antiwebs having this property, for every integerk≥1.

Keywords: (k-fold) graph coloring, (fractional) chromatic number, clique and stable set numbers, web and antiweb

1. Introduction

For any integers k ≥ 1 and x≥ 1, a k-fold x-coloring of a graph is an assignment of (at least)k distinct colors to each vertex from the set{1,2, . . . , x} such that any two adjacent vertices are assigned disjoint sets of colors [1, 2]. Each color used in the coloring defines what is called a stable set of the graph, i.e. a subset of pairwise nonadjacent vertices. We say that a graph G is k-fold x-colorable if G admits a k-fold x-coloring.

The smallest number x such that a graph G is k-fold x-colorable is called the k-th chromatic number ofG and is denoted by χ_k(G) [2]. Obviously, χ₁(G) = χ(G) is the conventional chromatic number of G. This variant of the conventional graph coloring

IA short version of this paper was presented atSimp´osio Brasileiro de Pesquisa Operacional, 2011.

This work is partially supported by a CNPq/FUNCAP Pronem project.

∗Corresponding author

Email addresses: mcampelo@lia.ufc.br(Manoel Campˆelo),correa@lia.ufc.br(Ricardo C.

Corrˆea),phablo@ime.usp.br(Phablo F. S. Moura),marciocs5@lia.ufc.br(Marcio C. Santos)

1Partially supported by CNPq-Brazil.

2Most of this work was done while the author was affiliated to Universidade Federal do Cear´a.

3Partially supported by Capes-Brazil.

Preprint submitted to Elsevier July 4, 2012

was introduced in the context of radio frequency assignment problem [3, 4]. Other applications include scheduling problems, bandwidth allocation in radio networks, fleet maintenance and traffic phasing problems [5–8].

Let n and p be integers such that p ≥ 1 and n ≥ 2p. As defined by Trotter, the webW_pⁿ is the graph whose vertices can be labelled as{v0, v1, . . . , v_n−1} in such a way that its edge set is{vivj | p≤ |i−j| ≤ n−p} [9]. The antiweb Wⁿ_p is defined as the complement of W_pⁿ. Examples are depicted in Figure 1, where the vertices are named according to an appropriate labelling (for the sake of convenience, we often name the vertices in this way in the remaining of the text). We observe that these definitions are interchanged in some references (see [10, 11], for instance). Webs and antiwebs form subclasses of circulant graphs that play an important role in the context of stable sets and vertex coloring problems [10–18]. Determining 1-fold coloring of circulant graphs is NP-Hard [19]. Therefore, determining chromatic parameters of subclasses of circulant graphs is worthwhile (see, for instance, [20] and references therein).

(a)W₃⁸ (b)W⁸₃

Figure 1: Example of a web and an antiweb.

In this paper, we derive a closed formula for thek-th chromatic number of webs and antiwebs. More specifically, we prove that χ_k(W_pⁿ) =l

kn p

mand χ_k(Wⁿ_p) =

kn

bⁿpc

, for every k ∈ N, thus generalizing similar results for odd cycles [2]. The denominator of each of these formulas is the size of the largest stable set in the corresponding graph,i.e.

thestability numberof the graph [9]. Indeed, it does not seem surprising that thek-th chromatic number is equal to that value. This claim comes from the idea of, starting with the circular numbering of the vertices, choosing a maximum independent set to describe a color, and shifting the class color appropriately around the circle. However, proving that this process works is not so simple, particularly in the case of antiwebs. Actually, this strategy does not work for general circulant graphs.

Besides its direct relation with the stability number, we also relate the k-th chro- matic number of webs and antiwebs with other parameters of the graph, such as the clique, chromatic and fractional chromatic numbers. Particularly, we derive necessary and sufficient conditions under which the classical bounds given by these parameters are tight.

In addition to the value of k-th chromatic number, we also provide optimal k-fold 2

colorings of W_pⁿ and Wⁿ_p. Based on the optimal colorings, we analyze when webs and antiwebs are critical with respect to this parameter. A graphGis said to be χ-critical ifχ(G−v) < χ(G), for allv ∈ V(G). An immediate consequence of this definition is that ifv is a vertex of aχ-critical graph G, then there exists an optimal 1-fold coloring ofG such that the color of v is not assigned to any other vertex. Not surprisingly, χ- critical subgraphs ofGplay an important role in several algorithmic approaches to vertex coloring. For instance, they are the core of the reduction procedures of the heuristic of [21] as well as they give facet-inducing inequalities of vertex coloring polytopes explored in cutting-plane methods [22–24]. From this algorithmic point of view, odd holes and odd anti-holes are (along with cliques) the most widely usedχ-critical subgraphs. It is already been noted that not only odd holes or odd anti-holes, but alsoχ-critical webs and antiwebs give facet-defining inequalities [18, 22].

We extend the concept ofχ-critical graphs toχ_k-critical graphs in a straightforward way. Then, we characterizeχ_k-critical webs and antiwebs, for any integer k ≥1. The characterization crucially depends on the greatest common divisors betweennandpand betweennand the stability number (which are equal for webs but may be different for antiwebs). Using the B´ezout’s identity, we show that there exists k ≥1 such thatW_pⁿ is χk-critical if, and only if, gcd(n, p) = 1. Moreover, when this condition holds, we determine all values ofkfor whichW_pⁿ isχk-critical. Similar results are derived forWⁿ_p, where the condition gcd(n, p) = 1 is replaced by gcd(n, p)6= p. As a consequence, we obtain that a web or an antiweb isχ-critical if, and only if, the stability number divides n−1. Such a characterization is trivial for webs but it was still not known for antiwebs [18]. More surprising, we show that beingχ-critical is also a sufficient condition for a web or an antiweb to beχk-critical for allk≥1.

Throughout this paper, we mostly use notation and definitions consistent with what is generally accepted in graph theory. Even though, let us set the grounds for all the notation used from here on. Given a graphG,V(G) andE(G) stand for its set of vertices and edges, respectively. The simplified notationV andE is prefered when the graphG is clear by the context. The complement ofGis written asG= (V, E). The edge defined by verticesuandv is denoted byuv.

As already mentioned, a set S ⊆V(G) is said to be astable set if all vertices in it are pairwise non-adjacent inG, i.e. uv6∈E∀u, v∈S. Thestability numberα(G) ofGis the size of the largest stable set ofG. Conversely, acliqueofGis a subsetK⊆V(G) of pairwise adjacent vertices. Theclique number ofG is the size of the largest clique and is denoted byω(G). For the ease of expression, we frequently refer to the graph itself as being a clique (resp. stable set) if its vertex set is a clique (resp. stable set). The fractional chromatic number of G, to be denoted ¯χ(G), is the infimum of ^x_k among the k-foldx-colorings [25]. It is known thatω(G)≤χ(G)¯ ≤χ(G) and _α(G)ⁿ ≤χ(G) [25]. A¯ graphGisperfectifω(H) =χ(H), for all induced subgraphH ofG.

A chordless cycleof length n is a graph Gsuch that V ={v1, v2, . . . , vn} and E = {vivi+1:i= 1,2, . . . , n−1} ∪ {v1vn}. Aholeis a chordless cycle of length at least four.

Anantiholeis the complement of a hole. Holes and antiholes are odd or even according to the parity of their number of vertices. Odd holes and odd antiholes are minimally imperfect graphs [26]. Observe that the odd holes and odd anti-holes are exactly the websW<sub>`</sub><sup>2`+1</sup>andW₂<sup>2`+1</sup>, for some integer`≥2, whereas the cliques are exactly the webs W₁ⁿ.

3

In the next section, we present general lower and upper bounds for thek-th chromatic number of an arbitrary simple graph. The exact value of this parameter is calculated for webs (Subsection 3.1) and antiwebs (Subsection 3.2). Some consequences of this result are presented in the following sections. In Section 4, we relate thek-th chromatic number of webs and antiwebs to their clique, integer and fractional chromatic numbers.

In particular, we identify which webs and antiwebs achieve the bounds given in Section 2 and those for which these bounds are strict. The definitions ofχk-critical andχ∗-critical graphs are introduced in Section 5, as a natural extension of the concept of χ-critical graphs. Then, we identify all webs and antiwebs that have these two properties.

2. Bounds for thek-th chromatic number of a graph

Two simple observations lead to lower and upper bounds for the k-th chromatic number of a graphG. On one hand, every vertex of a clique ofGmust receivek colors different from any color assigned to the other vertices of the clique. On the other hand, a k-fold coloring can be obtained by just replicating an 1-fold coloringktimes. Therefore, we get the following bounds which are tight, for instance, for perfect graphs.

Fact 1. For every k∈N,ω(G)≤χ(G)¯ ≤^χk_k^(G) ≤χ(G).

Another lower bound is related to the stability number, as follows. The lexicographic product of a graphGby a graphH is the graph that we obtain by replacing each vertex ofGby a copy ofH and adding all edges between two copies ofH if and only if the two replaced vertices ofGwere adjacent. More formally, thelexicographic product G◦H is a graph such that:

1. the vertex set ofG◦H is the cartesian productV(G)×V(H); and

2. any two vertices (u,u) and (v,ˆ ˆv) are adjacent in G◦H if and only if either uis adjacent tov, oru=vand ˆuis adjacent to ˆv

As noted by Stahl, another way to interpret thek-th chromatic number of a graphGis in terms ofχ(G◦Kk), whereKk is a clique with kvertices [2]. It is easy to see that a k-foldx-coloring ofGis equivalent to a 1-fold coloring ofG◦Kkwithxcolors. Therefore, χk(G) =χ(G◦Kk). Using this equation we can trivially derive the following lower bound for thek-th chromatic number of any graph.

Fact 2. For every graphGand every k∈N,χ_k(G)≥l

kn α(G)

m .

Proof. IfH1 andH2are two graphs, then α(H1◦H2) =α(H1)α(H2) [27]. Therefore, α(G◦Kk) =α(G)α(Kk) =α(G). We getχk(G) =χ(G◦Kk)≥l

kn α(G◦Kk)

m

=l

kn α(G)

m . Next we will show that the lower bound given by Fact 2 is tight for two classes of graphs, namely webs and antiwebs. Moreover, some graphs in these classes also achieve the lower and upper bounds stated by Fact 1.

4

3. The k-th chromatic number of webs and antiwebs

In the remaining, letnandpbe integers such thatp≥1 andn≥2pand let⊕stand for addition modulusn, i.e. i⊕j = (i+j) modnfori, j ∈Z. Let Nstand for the set of natural numbers (0 excluded). The following known results will be used later.

Lemma 1 (Trotter [9]). α(Wⁿ_p) =ω(W_pⁿ) =j

n p

k

andα(W_pⁿ) =ω(Wⁿ_p) =p.

Lemma 2 (Trotter [9]). Let n⁰ and p⁰ be integers such thatp⁰ ≥1 andn⁰ ≥2p⁰. The webW_pⁿ0⁰ is a subgraph of W_pⁿ if, and only if, np⁰≥n⁰pandn(p⁰−1)≤n⁰(p−1).

3.1. Web

We start by defining some stable sets of W_pⁿ. For each integer i ≥ 0, define the following sequence of integers:

S_i=hi⊕0, i⊕1, . . . , i⊕(p−1)i (1) Lemma 3. For every integeri≥0,Si indexes a maximum stable set of W_pⁿ.

Proof. By the symmetry ofW_pⁿ, it suffices to consider the sequenceS0. Letj1and j2

be inS0. Notice that|j1−j2| ≤p−1< p. Then,vj1vj2 ∈/ E(W_pⁿ), which proves thatS0

indexes an independent set with cardinalityp=α(W_pⁿ).

Using the above lemma and the sets S_i, we can now calculate the k-th chromatic number ofW_pⁿ. The main idea is to build a cover of the graph by stable sets in which each vertex ofW_pⁿ is covered at leastktimes.

Theorem 1. For every k∈N,χ_k(W_pⁿ) =l

kn p

m

=l

kn α(W_pⁿ)

m .

Proof. By Fact 2, we only have to show thatχ_k(W_pⁿ)≤l

kn p

m

, for an arbitraryk∈N. For this purpose, we show that Ξ(k) =hS0, Sp, . . . , S_(x−1)pi gives ak-fold x-coloring of W_pⁿ, withx=l

kn p

m

. We have that

Ξ(k) =

*

0⊕0,0⊕1, . . . ,0⊕p−1

| {z }

S0

, p⊕0, . . . , p⊕(p−1)

| {z }

Sp

, . . . ,

(x−1)p⊕0, . . . ,(x−1)p⊕(p−1)

| {z }

S_(x−1)p

+ .

Since the first element ofS<sub>(`+1)p</sub>, 0≤` < x−1, is the last element ofS`pplus 1 (modulus n), we have that Ξ(k) is a sequence (modulusn) of integer numbers starting at 0. Also, it hasl

kn p

m

p≥kn elements. Therefore, each element between 0 andn−1 appears at leastk times in Ξ(k). By Lemma 3, this means that Ξ(k) gives ak-fold l

kn p

m

-coloring

ofW_pⁿ, as desired.

5

3.2. Antiweb

As before, we proceed by determining stable sets of Wⁿ_p that cover each vertex at leastktimes. Now, we need to be more judicious in the choice of the stable sets ofWⁿ_p. We start by defining the following sequences (illustrated in Figure 2):

S₀ =

t ⁿ

α(Wⁿ_p)

: t= 0,1, . . . , α(Wⁿ_p)−1

(2) Si = hj⊕1 : j∈S_i−1i, i∈N

= hj⊕i: j∈S₀i, i∈N.

We claim that eachS_iindexes a maximum stable set ofWⁿ_p. This will be shown with the help of the following lemma.

Lemma 4. If x, y∈Randx≥y, thenbx−yc ≤ dxe − dye ≤ dx−ye.

Proof. It is clear thatx− dxe ≤0 anddye −y <1. By summing up these inequalities, we getx−y+dye − dxe<1. Therefore,bx−yc+dye − dxe ≤0 andbx−yc ≤ dxe − dye.

To get the second inequality, recall thatdx−ye+dye ≥ dx−y+ye=dxe.

We now get the counterpart of Lemma 3 for antiwebs.

Lemma 5. For every integeri≥0,S_i indexes a maximum stable set of Wⁿ_p.

Proof. By the symmetry of an antiweb and the definition of the S_i’s, it suffices to show the claimed result for S₀. Let j₁ and j₂ belong to S₀. We have to show that p≤ |j₁−j₂| ≤n−p. For the upper bound, note that

|j1−j2| ≤

&

(α(Wⁿ_p)−1)n α(Wⁿ_p)

'

=

&

n− n α(Wⁿ_p)

'

≤n−p,

where the last inequality is due toα(Wⁿ_p) =bn/pc. On the other hand,

|j1−j2| ≥min

`≥1

&

`n α(Wⁿ_p)

'

−

&

(`−1)n α(Wⁿ_p)

'!

.

By Lemma 4 andα(Wⁿ_p) =bn/pc, it follows that|j1−j2| ≥p. Therefore,S0indexes an

independent set of cardinalityα(Wⁿ_p).

The above lemma is the basis to give the expression of χk(Wⁿ_p). We proceed by choosing an appropriate family ofS_i’s and, then, we show that it covers each vertex at leastk times. We first consider the case wherek≤α(Wⁿ_p).

Lemma 6. Let be given positive integersn,p, andk≤α(Wⁿ_p). The index of each vertex ofWⁿ_p belongs to at leastkof the sequences S₀, S₁, . . . , S_x(k)−1, wherex(k) =

kn α(Wⁿ_p)

.

6

Proof. Let ` ∈ {1,2, . . . , k} and t ∈ {0,1, . . . , α(W<sup>n</sup><sub>p</sub>)−1}. Define A(`, t) as the se- quence comprising the (t+ 1)-th elements ofS0, S1, . . . , S_x(`)−1, that is,

A(`, t) =

t ⁿ

α(Wⁿ_p)

⊕i: i= 0,1, . . . ,

`n α(Wⁿ_p)

−1

.

Since`≤α(W<sup>n</sup><sub>p</sub>),A(`, t) has

`n α(Wⁿ_p)

distinct elements. Figure 2 illustrates these sets forW¹⁰₃ .

LetB(`, t) be the subsequence of A(`, t) formed by its first

(`+t)n α(Wⁿ_p)

−

tn α(Wⁿ_p)

≤

`n α(Wⁿ_p)

elements (the inequality comes from Lemma 4). In Figure 2(b),B(1, t) relates to the numbers in blue whereasB(2, t) comprises the numbers in blue and red. Notice that B(`, t) comprises consecutive integers (modulus n), starting at

tn α(Wⁿ_p)

⊕0 and ending at

(`+t)n α(Wⁿ_p)

⊕(−1). Consequently,B(`, t)⊆B(`+ 1, t).

Let C(1, t) =B(1, t) andC(`+ 1, t) =B(`+ 1, t)\B(`, t), for` < k. Similarly to B(`, t),C(`, t) comprises consecutive integers (modulusn), starting at

(`+t−1)n α(Wⁿ_p)

⊕0 and ending at

(`+t)n α(Wⁿ_p)

⊕(−1). Observe that the first element ofC(`, t+1) is the last element

ofC(`, t) plus 1 (modulus n). Then, C(`) = hC(`,0), C(`,1), . . . , C(`, α(W<sup>n</sup><sub>p</sub>)−1)iis a sequence of consecutive integers (modulusn) starting at the first element ofC(`,0), that is

(`−1)n α(Wⁿ_p)

⊕0, and ending at the last element ofC(`, α(W<sup>n</sup><sub>p</sub>)−1), that is (α(W<sup>n</sup><sub>p</sub>)+`−1)n

α(Wⁿ_p)

⊕(−1) =

(`−1)n α(Wⁿ_p)

⊕(−1).

This means thatC(`)≡ h0,1, . . . , n−1i. Therefore, for each`= 1,2, . . . , k,C(`) covers every vertex once. In addition, by definition ofC(`, t), ` > 1, the sequences Si’s from which the elements ofC(`, t) are taken are different from those ofC(1, t), C(2, t), . . . , C(`−

1, t). Consequently, every vertex is covered k times by C(1), C(2), . . . , C(k), and so is covered at leastk times byS0, S1, . . . , S_x(k)−1.

Now we are ready to prove our main result for antiwebs of this section.

Theorem 2. For every k∈N,χ_k(Wⁿ_p) =

kn α(Wⁿ_p)

.

Proof. By Fact 2, we only need to show the inequality χk(Wⁿ_p) ≤

kn α(Wⁿ_p)

. Let us write k=`α(W<sup>n</sup><sub>p</sub>) +i, for integers` ≥0 and 0≤i < α(Wⁿ_p). By lemmas 5 and 6, it is straightforward that the stable setsS₀, S₁, . . . , S_x−1, where x=

in α(Wⁿ_p)

, induce an 7

(a)W¹⁰₃ .

S0 S1 S2 S3 S4 S5 S6 S7

A(`,0) 0 1 2 3 4 5 6 · · · A(`,1) 4 5 6 7 8 9 0 · · · A(`,2) 7 8 9 0 1 2 3 · · ·

`= 1

`= 2

(b)C(1) in blue,C(2) in red.

Figure 2: Example of a 2-fold 7-coloring ofW¹⁰₃ . Recall thatα(W¹⁰₃ ) = 3.

i-fold x-coloring of Wⁿ_p. The same lemmas also give an α(Wⁿ_p)-fold n-coloring via sets S0, . . . , S_n−1. One copy of the first coloring together with`copies of the second one yield ak-fold coloring with`n+

in α(Wⁿ_p)

=

kn α(Wⁿ_p)

colors.

4. Relation with other parameters

The strict relationship between χk(G) and α(G) established for webs (Theorem 1) and anti-webs (Theorem 2) naturally motivates a similar question with respect to other parameters ofGknown to be related to the chromatic number. Particularly, we determine in this section when the bounds presented in Fact 1 are tight or strict. In what follows,

“a modb” stands for the remainder of the division ofabyb.

Proposition 1. Let GbeW_pⁿ orWⁿ_p andk∈N. Then,χ_k(G) =kχ(G)if, and only if, gcd(n, α(G)) =α(G)or k < _α(G)−r^α(G) , wherer=n modα(G).

Proof. By theorems 1 and 2,χk(G) =kχ(G) if, and only if,l

kn α(G)

m

=kl

n α(G)

m , which is also equivalent tol

kr α(G)

m

=kl

r α(G)

m

. This equality trivially holds if r= 0, that is, gcd(n, α(G)) = α(G). In the complementary case, l

r α(G)

m

= 1 and, consequently, the equality is equivalent to _α(G)^kr > k−1 or stillk < _α(G)−r^α(G) . Proposition 2. Let GbeW_pⁿ or Wⁿ_p andk∈N. Then,χk(G) =kω(G)if, and only if, gcd(n, p) =p.

Proof. Lets=n modp. Using Lemma 1, note thatn=bn/pcp+s=ω(G)α(G) +s.

By theorems 1 and 2, we get χ_k(G) =

kn α(G)

=kω(G) + ks

α(G)

.

The result then follows from the fact thats= 0 if, and only if, gcd(n, p) =p.

8

As we can infer from Lemma 1, ifpdividesn, then so doesα(W_pⁿ) andα(Wⁿ_p). Under such a condition, which holds for all perfect and some non-perfect webs and antiwebs, the lower and upper bounds given in Fact 1 are equal.

Corollary 1. Let Gbe W_pⁿ orWⁿ_p andk∈N. Then, kω(G) =χk(G) =kχ(G)if, and only if,gcd(n, p) =p.

On the other hand, the same bounds are always strict for some webs and antiwebs, including the minimally imperfect graphs.

Corollary 2. Let G be W_pⁿ or Wⁿ_p. If gcd(n−1, α(G)) = α(G) and α(G) > 1, then χ_k(G)< kχ(G), for all k >1. Moreover, if gcd(n−1, p) =pand p >1, then kω(G)<

χ_k(G)< kχ(G), for allk >1.

Proof. Assume that gcd(n−1, α(G)) =α(G) andα(G)≥2. Then,r:=n modα(G) = 1 and _α(G)−r^α(G) ≤2. By Proposition 1, χk(G)< kχ(G) for allk >1. To show the other inequality, assume that gcd(n−1, p) = pand p > 1. Then, gcd(n, p) 6=p. Moreover, α(W_pⁿ) =p > 1 and α(Wⁿ_p) = ⁿ⁻¹_p >1 so that gcd(n−1, α(G)) =α(G)>1. By the first part of this corollary and Proposition 2, the result follows.

To conclude this section, we relate the fractional chromatic number and the k-th chromatic number. By definition, for any graphG, these parameters are connected as follows:

¯

χ(G) = inf

χk(G)

k | k∈N

.

By theorems 1 and 2, ^χk_k^(G) ≥ _α(G)ⁿ , for every k ∈N, and this bound is attained with k=α(G). This leads to

Proposition 3. If GisW_pⁿ orWⁿ_p, then χ(G) =¯ _α(G)ⁿ .

Actually, the above expression holds for a larger class of graphs, namely vertex transi- tive graphs [25]. The following property readily follows in the case of webs and antiwebs.

Proposition 4. Let GbeW_pⁿ orWⁿ_p andk∈N. Then,χ_k(G) =kχ(G)¯ if, and only if,

kgcd(n,α(G)) α(G) ∈Z.

Proof. Let α = α(G) and g = gcd(n, α). By theorems 1 and 2 and Proposition 3, χ_k(G) =kχ(G) if, and only if,¯ ^kn_α ∈Z. Sincen/gandα/g are coprimes, ^kn_α = ^k(n/g)_α/g is

integer if, and only if, _α/g^k ∈Z.

By the above proposition, given any web or antiweb G such that α(G) does not dividen, there are always values ofksuch thatχk(G) =kχ(G) and values of¯ ksuch that χk(G)> kχ(G).¯

9

5. χk-critical web and antiwebs

We define a χ_k-critical graph as a graph G such thatχ_k(G−v) < χ_k(G), for all v∈V(G). If this relation holds for everyk ∈N, then Gis said to be χ_∗-critical. Now we investigate these properties for webs and antiwebs. The analysis is trivial in the case wherep= 1 because W₁ⁿ is a clique. For the case wherep > 1, the following property will be useful.

Lemma 7. If GisW_pⁿ orWⁿ_p andp >1, thenα(G−v) =α(G)andω(G−v) =ω(G), for allv∈V(G).

Proof. Letv ∈V(G). Sincep > 1,v is adjacent to some vertex u. Lemmas 3 and 5 imply that there is a maximum stable set ofGcontainingu. It follows that α(G−v) = α(G). Then, the other equality is a consequence ofα(G) =ω(G).

Additionally, the greatest common divisor betweenn and α(G) plays an important role in our analysis. For arbitrary nonzero integersaandb, the B´ezout’s identity guar- antees that the equationax+by= gcd(a, b) has an infinite number of integer solutions (x, y). As there always exist solutions with positivex, we can define

t(a, b) = min

t∈N: at−gcd(a, b)

b ∈Z

.

For our purposes, it is sufficient to consideraandb as positive integers.

Lemma 8. Let a, b ∈ N. If gcd(a, b) = b, then t(a, b) = 1. Otherwise, 0 < t(a, b) <

b gcd(a,b).

Proof. If gcd(a, b) =b, then we clearly havet(a, b) = 1. Now, assume that gcd(a, b)6=b.

Define the coprime integers a⁰ = a/gcd(a, b) and b⁰ = b/gcd(a, b)> 1. We have that t(a, b) =t(a⁰, b⁰) because gcd(a⁰, b⁰) = 1 and at−gcd(a,b)

b = ^a0t−1_b0 , for allt ∈ N. By the B´ezout’s identity, there are integers x > 0 and y such thata⁰x+b⁰y = 1. Take t =x modb⁰, that is, t=x−_x

b⁰

b⁰. Therefore, 0≤t < b⁰ and ^ta_b⁰⁻¹0 = −y−_x

b⁰

a⁰

∈Z. Actually,t >0 sinceb⁰>1. These properties oft imply that 0< t(a, b) =t(a⁰, b⁰)≤t <

b⁰.

5.1. Web

In this subsection, Theorem 1 is used to determine thek-chromatic number of the graph obtained by removing a vertex from W_pⁿ. For the ease of notation, along this subsection lett^?=t(n, p) =t(n, α(W_pⁿ)).

Lemma 9. For every k∈Nand every vertex v∈V(W_pⁿ),

χ_k(W_pⁿ−v) =









 lkn

p

m

, if gcd(n, p)6= 1, kn−bt?^kc

p

, if gcd(n, p) = 1,

10

Proof. Letq= gcd(n, p). First, suppose thatq >1. Using Lemma 2, we conclude that W_p/q^n/q is a proper subgraph of W_pⁿ. Then, by the symmetry of W_pⁿ, it is a subgraph of W_pⁿ−v. By Theorem 1, we have that

χk(W_pⁿ−v)≥

&_n

qk

p q

'

= nk

p

.

The converse inequality follows as a consequence ofχk(W_pⁿ−v)≤χk(W_pⁿ).

Now, assume thatq= 1.

Claim 1. χk(W_pⁿ−v)≤

nk−b_t?^kc

p

.

Proof. By the symmetry of W_pⁿ, we only need to prove the statement for v = v_n−1. Sinceq = 1, p divides nt^?−1. Let us use (1) to define Ξ =hS0, S_p, . . . , S(^nt?−1p −1)^pi, which is a sequence (modulusn) of integer numbers starting at 0 and ending at n−2.

Notice that it coverst^?times each integer from 0 ton−2. Using this sequencek t^?

times, we get a _k

t^?

t^?

-fold coloring ofW_pⁿ−v with ^nt?_p⁻¹_k

t^?

colors. If t^? divides k, then we are done. Otherwise, by Theorem 1 and the fact thatW_pⁿ−v⊆W_pⁿ, we can have an additional k−k

t^?

t^?

-fold coloring with at most l

n

p k−k t^?

t^?m

colors. Therefore, we obtain a k-fold coloring with at most ^nt?_p⁻¹k

t^?

+l

n

p k−k t^?

t^?m

=

nk−b_t?^kc

p

colors.

Claim 2. χk(W_pⁿ−v)≥l_(nt?−1)k pt^?

m

Proof. By Theorem 1, it suffices to show thatW_tⁿ?⁰ is a web included inW_pⁿ−v, where n⁰ = ^nt?_p⁻¹ ∈Zbecause q= 1. By Lemma 8, t^? < p implying thatn⁰ < n. Therefore, we only need to show thatW_tⁿ?⁰ is a subgraph ofW_pⁿ. First, notice that n≥2p+ 1 and so n⁰ ≥2t^?+^t?_p⁻¹ ≥2t^?. Thus,W_tⁿ?⁰ is indeed a web. To show that it is a subgraph of W_pⁿ, we apply Lemma 2. On one hand, nt^? ≥ nt^?−1 = n⁰p. On the other hand, n(t^?−1)≤n⁰(p−1) if, and only if,n⁰≤n−1. Therefore, the two conditions of Lemma 2

hold.

By claims 1 and 2, we get

&

nk−_k

t^?

p

'

≥χ(W_pⁿ−v)≥

&

nk−_t^k?

p '

.

To conclude the proof, we show that equality holds everywhere above. Let us write k=k

t^?

t^?+r, where 0≤r < t^?. By the definition oft^?, we have that ^nt?_p⁻¹ ∈Zbut

nr−1

p ∈/ Z. It follows that

&

nk−_t^k?

p '

≥

&

nk−_k

t^?

−1 p

'

=nt^?−1 p

k t^?

+

nr−1 p

= nt^?−1

p k

t^?

+nr p

=

&

nk−k t^?

p

' .

11

Remark 1. The proof of Lemma 9 provides the alternative equality χk(W_pⁿ−v) = l_kn−k

t?

p

m

when gcd(n, p) = 1.

Removing a vertex from a graph may decrease itsk-th chromatic number by a value varying from 0 tok. For webs, the expressions ofχ_k(W_pⁿ) andχ_k(W_pⁿ−v) given above together with Lemma 4 bound this decrease as follows.

Corollary 3. Letk∈Nandv∈V(W_pⁿ). Ifgcd(n, p)6= 1, thenχk(W_pⁿ) =χk(W_pⁿ−v).

Otherwise,j

k pt^?

k≤χk(W_pⁿ)−χk(W_pⁿ−v)≤l

k pt^?

m .

Remark 2. An important feature of aχ-critical graphGis that, for every vertex v∈ V(G), there is always an optimal coloring wherevdoes not share its color with the other vertices. Such a property makes it easier to show that inequalities based onχ-critical graphs are facet-defining for 1-fold coloring polytopes [18, 22, 24]. Fork≥2, Corollary 3 establishes that cliques are the unique webs for which there exists an optimal k-fold coloring where a vertex does not share any of itskcolors with the other vertices. Indeed, forp≥2 andk≥2, the upper bound given in Corollary 3 leads toχk(W_pⁿ)−χk(W_pⁿ−v)≤ k

2

< k.

Next, we identify the values ofn,p, andkfor which the lower bound given in Corol- lary 3 is nonzero. In other words, we characterize theχk-critical webs, for everyk∈N. Theorem 3. Let k ∈N. If gcd(n, p)6= 1, then W_pⁿ is not χk-critical. Otherwise, the following assertions are equivalent:

(i) W_pⁿ isχk-critical;

(ii) k≥pt^? or0< ^nk_p −j

nk p

k≤_pt^k?;

(iii) k≥pt^? ork=at^?+bp for some integersa≥1 andb≥0.

Proof. The first part is an immediate consequence of Corollary 3. For the second part, assume that gcd(n, p) = 1, which means that ^nt?_p⁻¹ ∈Z. Letr=kn modp, i.e.

r

p = ^kn_p −j

kn p

k

.So, assertion (ii) can be rewritten as

k≥pt^? ork≥rt^? withr >0. (3) On the other hand, by Theorem 1 and Remark 1, it follows that

χ_k(W_pⁿ) = kn

p

+ r

p

and χ_k(W_pⁿ−v) = kn

p

+

&

r−_t^k?

p '

.

Therefore, W_pⁿ is χk-critical if, and only if, l

r p

m

>l_r−k t?

p

m

. If r = 0, this means that l₋k

t?

p

m ≤ −1 or, equivalently, k ≥ pt^?. If r ≥ 1, then the condition is equivalent to 12

l_r−k t?

p

m ≤0 or still k≥ rt^?. As r < p, we can conclude thatW_pⁿ is χk-critical if, and only if, condition (3) holds.

To show that (3) implies assertion (iii), it suffices to show that k ≥rt^? and r >0 imply that there exist integersa≥1 andb ≥0 such thatk=at^?+bp. Indeed, notice that ^kn−r_p ∈Z. Then, ^knt?_p^−rt? = ^{(nt?−1)k+(k−rt}_p ^?) ∈Z. We can deduce that ^k−rt_p ^? ∈Z or, equivalently,k=rt^?+bpfor someb∈Z. Sincek≥rt^? andr≥1, the desired result follows.

Conversely, let us assume that k =at^?+bp for some integers a ≥1 and b ≥0. If a≥p, then we trivially get condition (3). So, assume thata < p. We claim thatr=a.

Indeed,

r= (nat^?) modp=nat^?−

(nt^?−1)a+a p

p=a− a

p

p=a.

Sincea≥1 andb≥0, we have that k≥rt^? andr >0.

As an immediate consequence of Theorem 3(iii), we have the characterization ofχ_∗- critical webs.

Theorem 4. The following assertions are equivalent:

(i) W_pⁿ isχ_∗-critical;

(ii) W_pⁿ isχ-critical;

(iii) α(W_pⁿ)divides n−1.

Proof. Since anyχ_∗-critical graph isχ-critical, we only need to prove that (ii) implies (iii), and (iii) implies (i). Moreover, (iii) is equivalently tot^? = gcd(n, p) = 1. To show the first implication, we apply Theorem 3(iii) withk= 1. It follows that gcd(n, p) = 1 andat^? ≤1 fora≥1. Therefore,t^? = gcd(n, p) = 1. For the second part, notice that anyk∈Ncan be written ask=at^?+bpfora=k≥1 andb= 0, whenevert^?= 1. The

result follows again by Theorem 3(iii).

Corollary 4. Cliques, odd holes and odd anti-holes are allχ_∗-critical.

5.2. Antiwebs

Now, we turn our attention toWⁿ_p. Similarly to the previous subsection, Theorem 2 is used to determine thek-chromatic number of the graph obtained by removing a vertex fromWⁿ_p. In this subsection, lett^?=t(n, α(Wⁿ_p)).

Lemma 10. For everyk∈Nand every vertex v∈V(Wⁿ_p),

χk(Wⁿ_p−v) =











kn α(Wⁿ_p)

if gcd(n, p) =p, k(n−1)

α(Wⁿ_p)

if gcd(n, p)6=p.

13

Proof. First assume thatpdivides n. Using Fact 1 and Corollary 1, we get kω(Wⁿ_p−v)≤χk(Wⁿ_p −v)≤χk(Wⁿ_p) =kω(Wⁿ_p).

By Lemma 7,ω(Wⁿ_p) =ω(Wⁿ_p−v) ifp >1. The same equality trivially holds whenp= 1 sinceWⁿ₁ has no edges. These facts and the above expression show thatχ_k(Wⁿ_p −v) = χ_k(Wⁿ_p) =

kn α(Wⁿ_p)

.

Now assume that gcd(n, p)6=p. Then, p >1 andn >2p. By Fact 2 and Lemma 7, we have that χk(Wⁿ_p −v) ≥

k(n−1) α(Wⁿ_p)

. Now, we claim thatWⁿ_p −v is a subgraph of Wⁿ⁻¹_p . First, notice that this antiweb is well-defined because n−1 ≥ 2p. Now, let v_iv_j ∈ E(Wⁿ_p −v) ⊂ E(Wⁿ_p). Then |i−j| < p or |i−j| > n−p > (n−1)−p.

Therefore, vivj ∈ E(Wⁿ⁻¹_p ). This proves the claim. Then, Theorem 2 implies that χk(Wⁿ_p −v)≤χk(Wⁿ⁻¹_p ) =

k(n−1) α(Wⁿ⁻¹_p )

. Moreover, since pdoes not divide n, it follows thatα(Wⁿ⁻¹_p ) =j

n−1 p

k=j

n p

k=α(Wⁿ_p). This shows the converse inequality χ_k(Wⁿ_p − v)≤

k(n−1) α(Wⁿ_p)

.

Using again Lemma 4, we can now bound the difference betweenχk(Wⁿ_p) andχk(Wⁿ_p− v).

Corollary 5. Let k∈Nandv ∈V(Wⁿ_p). Ifpdivides n, then χk(Wⁿ_p−v) =χk(Wⁿ_p).

Otherwise,

k α(Wⁿ_p)

≤χk(Wⁿ_p)−χk(Wⁿ_p−v)≤

k α(Wⁿ_p)

.

Remark 3. Fork≥2, no antiweb has an optimal k-fold coloring where a vertex does not share any of itskcolors with other vertices. Sinceα(Wⁿ_p)≥2, Corollary 5 establishes thatχk(Wⁿ_p)−χk(Wⁿ_p−v)≤k

2

< k, wheneverk≥2.

The above results also allow us to characterize theχk-critical antiwebs, as follows.

Theorem 5. Let k ∈N. If gcd(n, p) = p, then Wⁿ_p is not χk-critical. Otherwise, the following assertions are equivalent:

(i) Wⁿ_p isχ_k-critical;

(ii) k≥α(Wⁿ_p)or 0< ^nk

α(Wⁿ_p)−

nk α(Wⁿ_p)

≤ ^k

α(Wⁿ_p);

(iii) k ≥ α(Wⁿ_p) or k =at^?+bq for some integers a≥ 1 and b ≥ a(gcd(n,α(Wⁿ_p))−t^?)

q ,

whereq=α(Wⁿ_p)/gcd(n, α(Wⁿ_p)).

Proof. We use Theorem 2 and Lemma 10 to get the expressions of χ_k(Wⁿ_p) and χ_k(Wⁿ_p−v). Then, the first part of the statement immediately follows. Now assume that

14

gcd(n, p)6=p. Letα=α(Wⁿ_p) andr =kn modα so that _α^r = ^kn_α −kn α

.It follows that

χ_k(Wⁿ_p) = kn

α

+lr α m

and χ_k(Wⁿ_p −v) = kn

α

+ r−k

α

.

Therefore, Wⁿ_p is χk-critical if, and only if, _r

α

> _r−k

α

. If r = 0, this means that −_α^k

≤ −1 or, equivalently, k ≥ α. If r ≥ 1, then the condition is equivalent to _r−k

α

≤0 or stillk≥r. Asr < α, we can conclude that Wⁿ_p isχk-critical if, and only if,

k≥α, ork≥r andr >0. (4)

Notice that this is exactly assertion (ii).

To show the remaining equivalence, we use again (4). Let g = gcd(n, α). By the definitions of r and t^?, we have that ^gk−rt_α ^? = ^nk−r_α t^?− ^nt?_α^−gk ∈ Z. It follows that k=at^?+bq for some b∈Zand a=r/g∈Z. Therefore, the second alternative of (4) implies the second alternative of assertion (iii). This leads to one direction of the desired equivalence.

Conversely, let us assume that assertion (iii) holds, that is, there exist integersa≥1 and b such that k = at^?+bq and bq ≥ ag−at^?. Then, k ≥ ag. If ag ≥ α, then we trivially get item (ii). So, assume thatag < α. We will show thatr=ag. Indeed,

r= (nat^?+nb

g α) modα= (nat^?) modα= nat^?−

(nt^?−g)a+ag α

α=ag−jag α k

α=ag.

Sincea≥1 andk≥ag, we have thatk≥randr >0, showing the converse implication.

The counterpart of Theorem 4 for antiwebs can be stated now.

Theorem 6. The following assertions are equivalent:

(i) Wⁿ_p isχ∗-critical;

(ii) Wⁿ_p isχ-critical;

(iii) α(Wⁿ_p)divides n−1.

Proof. Letα=α(Wⁿ_p), g = gcd(n, α) andq =α/g. It is trivial that (i) implies (ii).

Now assume thatWⁿ_p isχ-critical. By applying Theorem 5(iii) withk= 1, we have that at^?+bq= 1 andbq≥ag−at^?, for some integersa≥1 andb. Then,ag≤1. It follows thata=g= 1 andb=^1−t_q^? ∈Z. Since gcd(n, p)6=p, due to Theorem 5, and 1≤t^?< q, due to Lemma 8, we obtain that 0≥b≥l

1−q q

m

= 0. Therefore,t^?=g= 1 showing that αdividesn−1.

Conversely, assume that ⁿ⁻¹_α ∈Z, i.e. t^? =g= 1. Then, α6= ⁿ_p, which implies that gcd(n, p)6=p. Moreover, any k ∈Ncan be written as k=at^?+bq fora=k≥1 and b= 0. Sinceaandbsatisfy the conditions of Theorem 5(iii),Wⁿ_p isχ_∗-critical.

15

Corollary 6. If pdivides n−1 thenW_pⁿ andWⁿ_p areχ_∗-critical.

Proof. By Theorem 4, it remains to show thatWⁿ_p isχ_∗-critical. Under the hypothesis, we have thatα(Wⁿ_p) = ⁿ⁻¹_p . Since ⁿ⁻¹

α(Wⁿ_p) =p∈Z, the result follows by Theorem 6.

[1] G. Ren, Y. Bu,k-fold coloring of planar graphs, Science China Mathematics 53 (10) (2010) 2791–

2800.

[2] S. Stahl, n-tuple colorings and associated graphs, Journal of Combinatorial Theory, Series B 20 (1976) 185–203.

[3] L. Narayanan, Channel Assignment and Graph Multi-coloring, Wiley, 2002.

[4] F. S. Roberts, On the mobile radio frequency assignment problem and the traffic light phasing problem, Annals of NY Academy of Sciences 319 (1979) 466–483.

[5] V. C. Barbosa, E. Gafni, Concurrency in heavily loaded neighborhood-constrained systems, ACM Transactions on Programming Languages and Systems 11 (1989) 562–584.

[6] M. Halld´orsson, G. Kortsarz, Tools for multicoloring with applications to planar graphs and partial k-trees, Journal of Algorithms 42 (2002) 334–366.

[7] R. Klasing, N. Morales, S. P´erennes, On the complexity of bandwidth allocation in radio networks, Theoretical Computer Science 406 (3) (2008) 225–239.

[8] R. J. Opsut, F. S. Roberts, On the fleet maintenance, mobile radio frequency, task assigment and traffic phasing problems, in: G. Chartrand, Y. Alavi, D. Goldsmith, L. Lesniak-Foster, D. Lick (Eds.), The Theory and Applications of Graphs, Wiley, 1981, pp. 479–492.

[9] L. E. Trotter, A class of facet producing graphs for vertex packing polyhedra, Discrete Mathematics 12 (1975) 373–388.

[10] A. Pˆecher, A. K. Wagler, Almost all webs are not rank-perfect, Mathematical Programming 105 (2006) 311–328.

[11] A. K. Wagler, Antiwebs are rank-perfect, 4OR: A Quarterly Journal of Operations Research 2 (2004) 149–152.

[12] E. Cheng, S. de Vries, Antiweb-wheel inequalities and their separation problems over the stable set polytopes, Mathematical Programming 92 (2002) 153–175.

[13] E. Cheng, S. de Vries, On the facet-inducing antiweb-wheel inequalities for stable set polytopes, SIAM Journal on Discrete Mathematics 15 (4) (2002) 470–487.

[14] F. Eisenbrand, G. Oriolo, G. Stauffer, P. Ventura, The stable set polytope of quasi-line graphs, Combinatorica 28 (2008) 45–67.

[15] A. Galluccio, A. Sassano, The rank facets of the stable set polytope for claw-free graphs, Journal of Combinatorial Theory, Series B 69 (1) (1997) 1–38.

[16] R. Giles, L. E. Trotter, On stable set polyhedra forK1,3-free graphs, Journal of Combinatorial Theory, Series B 31 (3) (1981) 313–326.

[17] G. Oriolo, G. Stauffer, Clique-circulants and the stable set polytope of fuzzy circular interval graphs, Mathematical Programming 115 (2008) 291–317.

[18] G. Palubeckis, Facet-inducing web and antiweb inequalities for the graph coloring polytope, Discrete Applied Mathematics 158 (2010) 2075–2080.

[19] B. Codenotti, I. Gerace, S. Vigna, Hardness results and spectral techniques for combinatorial prob- lems on circulant graphs, Linear Algebra Applications 285 (1998) 123–142.

[20] J. Barajas, O. Serra, On the chromatic number of circulant graphs, Discrete Mathematics 309 (2009) 5687–5696.

[21] F. Herrmann, A. Hertz, Finding the chromatic number by means of critical graphs, ACM Journal of Experimental Algorithmics 7 (2002) 1–9.

[22] M. Campˆelo, R. Corrˆea, Y. Frota, Cliques, holes and the vertex coloring polytope, Information Processing Letters 89 (4) (2004) 159–164.

[23] P. Hansen, M. Labb´e, D. Schindl, Set covering and packing formulations of graph coloring: Algo- rithms and first polyhedral results, Discrete Optimization 6 (2) (2009) 135–147.

[24] I. M´endez-D´ıaz, P. Zabala, A cutting plane algorithm for graph coloring, Discrete Applied Mathe- matics 156 (2) (2008) 159–179.

[25] E. R. Scheinerman, D. H. Ullman, Fractional Graph Theory: A Rational Approach to the Theory of Graphs, Wiley-Interscience, 1997.

[26] M. Chudnovsky, N. Robertson, P. Seymour, R. Thomas, The strong perfect graph theorem, Annals of Mathematics 164 (2006) 51–229.

16

[27] D. Geller, S. Stahl, The chromatic number and other functions of the lexicographic product, Journal of Combinatorial Theory, Series B 19 (1975) 87–95.

17