Optimal k-fold colorings of webs and antiwebs
A -fold -coloring of a graph is an assignment of (at least) distinct colors from the set to each vertex such that any two adjacent vertices are assigned disjoint sets of colors. The smallest number such that admits a -fold -coloring is the -th chromatic number of , denoted by . We determine the exact value of this parameter when is a web or an antiweb. Our results generalize the known corresponding results for odd cycles and imply necessary and sufficient conditions under which attains its lower and upper bounds based on the clique, the fractional chromatic and the chromatic numbers. Additionally, we extend the concept of -critical graphs to -critical graphs. We identify the webs and antiwebs having this property, for every integer .
Keywords: (-fold) graph coloring, (fractional) chromatic number, clique and stable set numbers, web and antiweb
For any integers and , a -fold -coloring of a graph is an assignment of (at least) distinct colors to each vertex from the set such that any two adjacent vertices are assigned disjoint sets of colors [20, 23]. 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 is -fold -colorable if admits a -fold -coloring. The smallest number such that a graph is -fold -colorable is called the -th chromatic number of and is denoted by . Obviously, is the conventional chromatic number of . This variant of the conventional graph coloring was introduced in the context of radio frequency assignment problem [15, 21]. Other applications include scheduling problems, bandwidth allocation in radio networks, fleet maintenance and traffic phasing problems [1, 10, 13, 16].
Let and be integers such that and . As defined by Trotter, the web is the graph whose vertices can be labelled as in such a way that its edge set is . The antiweb is defined as the complement of . Examples are depicted in Figure b, 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 [19, 25], for instance). Webs and antiwebs form a class of graphs that play an important role in the context of stable sets and vertex coloring problems [3, 4, 6, 7, 9, 17, 18, 19, 25].
In this paper, we derive a closed formula for the -th chromatic number of webs and antiwebs. More specifically, we prove that and , for every , thus generalizing similar results for odd cycles . The denominator of each of these formulas is the size of the largest stable set in the corresponding graph, i.e. the stability number of the graph . Besides this direct relation with the stability number, we also relate the -th chromatic 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 -th chromatic number, we also provide optimal -fold colorings of and . Based on the optimal colorings, we analyse when webs and antiwebs are critical with respect to this parameter. A graph is said to be -critical if , for all . An immediate consequence of this definition is that if is a vertex of a -critical graph , then there exists an optimal -fold coloring of such that the color of is not assigned to any other vertex. Not surprisingly, -critical subgraphs of play an important role in several algorithmic approaches to vertex coloring. For instance, they are the core of the reduction procedures of the heuristic of  as well as they give facet-inducing inequalities of vertex coloring polytopes explored in cutting-plane methods [2, 11, 14]. From this algorithmic point of view, odd holes and odd anti-holes are (along with cliques) the most widely used -critical subgraphs. It is already been noted that not only odd holes or odd anti-holes, but also -critical webs and antiwebs give facet-defining inequalities [2, 18].
We extend the concept of -critical graphs to -critical graphs in a straightforward way. Then, we characterize -critical webs and antiwebs, for any integer . The characterization crucially depends on the greatest common divisors between and and between and the stability number (which are equal for webs but may be different for antiwebs). Using the Bézout’s identity, we show that there exists such that is -critical if, and only if, . Moreover, when this condition holds, we determine all values of for which is -critical. Similar results are derived for , where the condition is replaced by . As a consequence, we obtain that a web or an antiweb is -critical if, and only if, the stability number divides . Such a characterization is trivial for webs but it was still not known for antiwebs . More surprising, we show that being -critical is also a sufficient for a web or an antiweb to be -critical for all .
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 graph , and stand for its set of vertices and edges, respectively. The simplified notation and is prefered when the graph is clear by the context. The complement of is written as . The edge defined by vertices and is denoted by .
As already mentioned, a set is said to be a stable set if all vertices in it are pairwise non-adjacent in , i.e. . The stability number of is the size of the largest stable set of . Conversely, a clique of is a subset of pairwise adjacent vertices. The clique number of is the size of the largest clique and is denoted by . 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 , to be denoted , is the infimum of among the -fold -colorings . It is known that and . A graph is perfect if , for all induced subgraph of .
A chordless cycle of length is a graph such that and . A hole is a chordless cycle of length at least four. An antihole is 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 . Observe that the odd holes and odd anti-holes are exactly the webs and , for some integer , whereas the cliques are exactly the webs .
In the next section, we present general lower and upper bounds for the -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 the -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 of -critical and -critical graphs are introduced in Section 5, as a natural extension of the concept of -critical graphs. Then, we identify all webs and antiwebs that have these two properties.
2 Bounds for the -th chromatic number of a graph
Two simple observations lead to lower and upper bounds for the -th chromatic number of a graph . On one hand, every vertex of a clique of must receive colors different from any color assigned to the other vertices of the clique. On the other hand, a -fold coloring can be obtained by just replicating an -fold coloring times. Therefore, we get the following bounds which are tight, for instance, for perfect graphs.
For every , .
Another lower bound is related to the stability number, as follows. The lexicographic product of a graph by a graph is the graph that we obtain by replacing each vertex of by a copy of and adding all edges between two copies of if and only if the two replaced vertices of were adjacent. More formally, the lexicographic product is a graph such that:
the vertex set of is the cartesian product ; and
any two vertices and are adjacent in if and only if either is adjacent to , or and is adjacent to
As noted by Stahl, another way to interpret the -th chromatic number of a graph is in terms of , where is a clique with vertices . It is easy to see that a -fold -coloring of is equivalent to a -fold coloring of with colors. Therefore, . Using this equation we can trivially derive the following lower bound for the -th chromatic number of any graph.
For every graph and every , .
Proof: If and are two graphs, then . Therefore, . We get .
3 The -th chromatic number of webs e antiwebs
In the remaining, let and be integers such that and and let stand for addition modulus , i.e. for . Let stand for the set of natural numbers ( excluded). The following known results will be used later.
Lemma 3 (Trotter )
Lemma 4 (Trotter )
Let and be integers such that and . The web is a subgraph of if, and only if, and .
We start by defining some stable sets of . For each integer , define the following sequence of integers:
For every integer , indexes a maximum stable set of .
Proof: By the symmetry of , it suffices to consider the sequence . Let and be in . Notice that . Then, , which proves that indexes an independent set with cardinality .
Using the above lemma and the sets , we can now calculate the -th chromatic number of . The main ideia is to build a cover of the graph by stable sets in which each vertex of is covered at least times.
For every , .
Proof: By Lemma 2, we only have to show that , for an arbitrary . For this purpose, we show that gives a -fold -coloring of , with . We have that
Since the first element of , , is the last element of plus (modulus ), we have that is a sequence (modulus ) of integer numbers starting at . Also, it has elements. Therefore, each element between and appears at least times in . By Lemma 5, this means that gives a -fold -coloring of , as desired.
As before, we proceed by determining stable sets of that cover each vertex at least times. Now, we need to be more judicious in the choice of the stable sets of . We start by defining the following sequences (illustrated in Figure 2):
We claim that each indexes a maximum stable set of . This will be shown with the help of the following lemmas.
If and , then .
Proof: It is clear that and . By summing up these inequalities, we get . Therefore, . To get the second inequality, recall that .
For every antiweb and every integer , .
Proof: Since , we have that , which implies . Since is integer, the result follows.
For and every integer , .
We now get the counterpart of Lemma 5 for antiwebs.
For every integer , indexes a maximum stable set of .
Proof: By the symmetry of an antiweb and the definition of the ’s, it suffices to show the claimed result for . Let and belong to . We have to show that . For the upper bound, note that
Lemma 7 implies that this last term is no more than , that is, . On the other hand,
By Lemma 8, it follows that . Therefore, indexes an independent set of cardinality .
The above lemma is the basis to give the expression of . We proceed by choosing an appropriate family of ’s and, then, we show that it covers each vertex at least times. We first consider the case where .
Let be given positive integers , , and . The index of each vertex of belongs to at least of the sequences , where .
Proof: Let and . Define as the sequence comprising the -th elements of , that is,
Since , has distinct elements. Figure 2 illustrates these sets for .
Let be the subsequence of formed by its first elements (the inequality comes from Lemma 6). In Figure b, relates to the numbers in blue whereas comprises the numbers in blue and red. Notice that comprises consecutive integers (modulus ), starting at and ending at . Consequently, .
Let and , for . Similarly to , comprises consecutive integers (modulus ), starting at and ending at . Observe that the first element of is the last element of plus (modulus ). Then, is a sequence of consecutive integers (modulus ) starting at the first element of , that is , and ending at the last element of , that is
This means that . Therefore, for each , covers every vertex once. Consequently, every vertex is covered times by , and so is covered at least times by .
Now we are ready to prove our main result for antiwebs.
For every , .
Proof: By Lemma 2, we only need to show the inequality . Let us write , for integers and . By lemmas 9 and 10, it is straightforward that the stable sets , where , induce an -fold -coloring of . The same lemmas also give an -fold -coloring via sets . One copy of the first coloring together with copies of the second one yield a -fold coloring with colors.
4 Relation with other parameters
The strict relationship between and established for webs (Theorem 1) and anti-webs (Theorem 2) naturally motivates a similar question with respect to other parameters of known to be related to the chromatic number. Particularly, we determine in this section when the bounds presented in Lemma 1 are tight or strict.
Let be or and . Then, if, and only if, or , where .
Proof: By theorems 1 and 2, if, and only if, , which is also equivalent to . This equality trivially holds if , that is, . In the complementary case, and, consequently, the equality is equivalent to or still .
Let be or and . Then, if, and only if, .
The result then follows from the fact that if, and only if, .
As we can infer from Lemma 3, if divides , then so does and . Under such a condition, which holds for all perfect and some non-perfect webs and antiwebs, the lower and upper bounds given in Lemma 1 are equal.
Let be or and . Then, if, and only if, .
On the other hand, the same bounds are always strict for some webs and antiwebs, including the minimally imperfect graphs.
Let be or . If and , then , for all . Moreover, if and , then , for all .
Proof: Assume that and . Then, and . By Proposition 1, for all . To show the other inequality, assume that and . Then, . Moreover, and so that . 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 -th chromatic number. By definition, for any graph , these parameters are connected as follows:
If is or , then .
Actually, the above expression holds for a larger class of graphs, namely vertex transitive graphs . The following property readily follows in the case of webs and antiwebs.
Let be or and . Then, if, and only if, .
By the above proposition, given any web or antiweb such that does not divide , there are always values of such that and values of such that .
5 -critical web and antiwebs
We define a -critical graph as a graph such that , for all . If this relation holds for every , then is said to be -critical. Now we investigate these properties for webs and antiwebs. The analysis is trivial in the case where because is a clique. For the case where , the following property will be useful.
If is or and , then and , for all .
Additionally, the greatest common divisor between and plays an important role in our analysis. For arbitrary nonzero integers and , the Bézout’s identity guarantees that the equation has an infinity number of integer solutions . As there always exist solutions with positive , we can define
For our purposes, it is sufficient to consider and as positive integers.
Let . If , then . Otherwise, .
Proof: If , then we clearly have . Now, assume that . Define the coprime integers and . We have that because and , for all . By the Bézout’s identity, there are integers and such that . Take , that is, . Therefore, and . Actually, since . These properties of imply that .
In this subsection, Theorem 1 is used to determine the -chromatic number of the graph obtained by removing a vertex from . For the ease of notation, along this subsection let .
For every and every vertex ,
The converse inequality follows as a consequence of .
Now, assume that .
Proof: By the symmetry of , we only need to prove the statement for . Since , divides . Let us use (1) to define , which is a sequence (modulus ) of integer numbers starting at and ending at . Notice that it covers times each integer from to . Using this sequence times, we get a -fold coloring of with colors. If divides , then we are done. Otherwise, by Theorem 1 and the fact that , we can have an additional -fold coloring with at most colors. Therefore, we obtain a -fold coloring with at most colors.
Proof: By Theorem 1, it suffices to show that is a web included in , where because . By Lemma 12, implying that . Therefore, we only need to show that is a subgraph of . First, notice that and so . Thus, is indeed a web. To show that it is a subgraph of , we apply Lemma 4. On one hand, . On the other hand, if, and only if, . Therefore, the two conditions of Lemma 4 hold.
To conclude the proof, we show that equality holds everywhere above. Let us write , where . By the definition of , we have that but . It follows that
The proof of Lemma 13 provides the alternative equality when .
Removing a vertex from a graph may decrease its -th chromatic number of a value varying from to . For webs, the expressions of and given above together with Lemma 6 bound this decrease as follows.
Let and . If , then