Let be a simple connected graph with vertex set and edge set . A -total coloring of is an assignment of colors to its elements (vertices and edges) so that adjacent or incident elements have distinct colors. The total chromatic number is the smallest integer for which has a -total coloring. Clearly, , where is the maximum degree of , and the Total Coloring Conjecture (TCC), posed fifty years ago independently by Vizing vizing and Behzad et al. Behzad , states that . Graphs with are said to be Type and graphs with are said to be Type . In 1977, Kostochka kostochka77 verified the TCC for all graphs with maximum degree 4, but the TCC has not been settled for all regular graphs, where all vertices have the same degree. Although the TCC is trivially settled for all bipartite graphs, the problem of determining the total chromatic number of a -regular bipartite graph is NP-hard, for each fixed ColinSanchez , exposing how challenging the problem of total coloring is.
The direct product (also called tensor product or categorical product) of two graphs and is a graph denoted by , whose vertex set is the Cartesian product of the vertex sets and that is , for which vertices and are adjacent if and only if and . The definition clearly implies that the maximum degree satisfies , and that the direct product is a regular graph if and only if both and are regular graphs. Concerning the category of graphs, where objects are graphs and morphisms are graph homomorphisms, we know that the direct product is the categorical product that is defined by projections and . The direct product has the commutative property, that is, the graph is isomorphic to the graph . Moreover, the direct product is bipartite if and only if or is bipartite, and it is disconnected if and only if and are bipartite graphs. In particular, in case both and are connected bipartite graphs, the direct product has exactly two bipartite connected components.
A total coloring defines a vertex coloring and an edge coloring, and both coloring problems have been studied with respect to the direct product. A -vertex (resp. edge) coloring of a graph is an assignment of colors to its vertices (resp. edges) so that adjacent vertices (resp. incident edges) have distinct colors. The chromatic number (resp. index) is the smallest integer for which a graph has a -vertex (resp. edge) coloring. Concerning vertex coloring, Hedetniemi conjectured in 1966 that the chromatic number of would be equal to the minimum of the chromatic numbers of and and recently, fifty years later, the conjecture has been refuted by Shitov Shitov . Concerning edge coloring, Jaradat jaradat proved that if one factor reaches the lower bound for edge coloring, so does the direct product.
A cycle graph, denoted by , is a connected 2-regular graph. The graph is Type 1 if is multiple of and Type 2, otherwise yapbook . The direct product of cycle graphs is a 4-regular graph, and it is disconnected precisely when both and are even in which case consists of two isomorphic 4-regular bipartite connected components each being a spanning subgraph of the complete bipartite graph .
Concerning the total coloring of the direct product, there are few known results. Most classified direct product of graphs are Type 1. Prnaver and Zmazek zmazek established the TCC for the direct product of a path of length greater or equal to 3 and an arbitrary graph with chromatic index . They additionally proved, for , that and are equal to 5. Recently, the total chromatic number of direct product of complete graphs has been fully determined as being Type 1 with the exception of Carol2021 .
An equitable total coloring is a total coloring where the number of elements colored with each color differs by at most one. In 2009, Tong et al. tong showed that the equitable total chromatic number of the Cartesian product between and , denoted by , is equal to 5 for all possible values . It is known that geethasum , therefore we know that , for all .
In 2018, Geetha and Somasundaram geethasum conjectured that, except for , all direct product of cycle graphs are Type 1. As evidence, they established three infinite families of Type 1 direct product of cycle graphs: for arbitrary , if , where and . In order to describe the claimed total colorings for the three infinite families, they present four tables whose entries are the 5 colors given to suitable matchings between independent sets of vertices that are colored with no conflicts.
In Section 2, we present a general pattern that gives a 5-total coloring for all graphs , except for . Therefore we ensure that the open remaining infinite families of are also Type 1. In Section 3, we investigate further conditions that ensure that the direct product is Type 1. We ask whether one factor reaching the lower bound is enough to ensure that the direct product also reaches the lower bound for the total chromatic number. We manage to classify into Type 1 or Type 2 additional bipartite direct product of graphs.
2 Total coloring of
In this section, we prove that the graph is Type 1, except for . Note that the graph is Type , as it is isomorphic to two copies of , well known to be Type 2, and it is the single exception among the direct product of cycle graphs .
The present section is devoted to the proof of Theorem 1.
Except for , the graph is Type 1.
We omit five particular cases that are too small to apply the used technique, but are easy to verify to be Type 1, for instance by using the free open-source mathematics software system Sage Math. They are:, , , and . Figure 1 presents a 5-total coloring of . Therefore, as is isomorphic to , we shall consider in our proof with and . We shall write , for and and . Note that as , the next case is for which the remainder of the division by 5 is 2. For instance, to obtain a 5-total coloring of , we consider the isomorphic graph and write , with and .
In order to prove Theorem 1, we construct a -total coloring of an auxiliary graph, called matching quotient, from which we obtain a -total coloring of . We use suitable independents sets and matchings between them, inspired by the strategy used by Geetha and Somasundaram geethasum to total color the three particular infinite families. For , denote by , and . Clearly, each set has elements, and sets and are two perfect matchings between independent sets and in . From that, we define the matching quotient of , denoted by , as the multigraph where each of its vertices correspond to an independent set , and we have two edges between and which correspond to and . Note that a 5-total coloring of the matching quotient represents a 5-total coloring of . Figure 2 presents an example of a matching quotient, by depicting the matching quotient of .
In Subsections 2.1 and 2.2, we establish a 5-total coloring of the matching quotient of , proving Theorem 1. In Subsection 2.1, we exhibit a 5-total coloring of the matching quotients of five base infinite families: , and , for . Note that the base infinite families are those where for and . We observe that the 5-total coloring of the base infinite family acts as a pattern. In Subsection 2.2, for the matching quotient of , with an arbitrary large value of , we observe that we can split this graph into (possibly many) pattern blocks that are identified with the matching quotient of , and one base block which is identified with the matching quotient of each base infinite family , , , and . The 5-total colorings of the matching quotients, given in Subsection 2.1, produce a 5-total coloring of each block such that there are no conflicts of colors. The strategy of splitting the graph into blocks gives a 5-total coloring of the matching quotient of , ensuring that is Type 1.
2.1 Base infinite families
We consider first the base infinite families with and . We refer to Figures 3, 4, 5, 6 and 7 for the 5-total coloring of each base case. Note that, the 5-total colorings of the base infinite families have important features in common: the same color 1 (pink) given to the vertex , the same color 2 (green) given to the matching and the same color 4 (yellow) given to the matching . These shared features provide the needed compatibility that allows us to define a common pattern used when we deal with larger values of .
2.2 Merging the pattern to generate a 5-total coloring for arbitrary
To obtain a 5-total coloring of the matching quotient for an arbitrary large value of , we repeatedly merge the pattern block given by 5-total coloring of the matching quotient with the 5-total coloring of its base block , for . Note that the colored is the first base case and is also the only pattern used for an arbitrary value of independently of its base case.
Recall that, as we argued in the beginning of Section 2, by swapping and , we are always able to consider and write a large value of as , for and . We optimally color first its base block and then repeatedly merge with copies of the optimally colored pattern block . So the 5-total coloring of is defined by two steps as follows:
Base step: For each , the color of (respectively, and ) in is the same as the color of (respectively, and ) in its base case .
Pattern step: For each , write , and the color of (respectively, and ) in is the same as the color of (respectively, and ) in the pattern .
For instance, consider and please refer to Figure 8. Note that, in the base step, we color the elements and , for , of with the same colors as its base infinite family . Now, in the pattern step, we color the elements and , for of with the same colors as the pattern (as in Figure 3). Analogously, when we merge the pattern twice into to obtain a 5-total coloring of the matching quotient as highlighted in Figure 9 by elements and , for . Thus, for a general we merge patterns into the corresponding base infinite family to obtain a 5-total coloring of the matching quotient .
Note that there is no conflict between the assigned colors in the defined 5-total coloring for an arbitrary value of . Indeed, we already know that each base infinite family has color 1 (pink) to , color 2 (green) to and color 4 (yellow) to . Note that, regardless of how many times we use the pattern, there is no conflict between the patterns, as the edges colored 2 (green) and 4 (yellow) in the case base are the ones used between the patterns. Also, there is no conflict between the base case and the pattern, as the edges colored 2 (green) and 4 (yellow) between and (pink), and between colored 1 (pink) and , are both used in our base cases, where both and have a colored 1 (pink) neighbor vertex.
3 On total coloring bipartite direct product of graphs
In this section, we propose and investigate two questions motivated by the search for a general pattern for the classification into Type 1 or Type 2 of the direct product of two graphs. In this sense, it is natural to seek for sufficient conditions for the direct product to be Type 1. Prnaver and Zmazek zmazek previously proved that if admits a -edge coloring, then , for , is Type 1. Mackeigan and Janssen JaMa19 subsequently proved that if is Type 1, then is also Type 1, for any bipartite graph . Recall that for edge coloring, Jaradat jaradat proved that if one factor reaches the lower bound for edge coloring, so does the direct product. We investigate whether an analogous property holds for total coloring:
Given a Type 1 graph and an arbitrary graph , is the direct product Type 1?
The analogous question has been considered for the Cartesian product, but has only been partially answered. It is known that if the factor with largest vertex degree is of Type 1, then the Cartesian product is also of Type 1 zmazek2002 . So far, all known Type 2 direct product of two graphs are the direct product , where and are Type 2, including cases with . The known Type 2 direct product of two graphs are: , , , and for not a multiple of 3. On the other hand, a Type 1 direct product of two graphs can be obtained when and are Type 1, when and are Type 2, or else when one of them is Type 1 and the other is Type 2. For instance, is Type 1 when
is odd and Type 2 whenis even, and yet the direct product is Type 1 when both and are odd Carol2021 , when are both even geethasum , or else when or is even JaMa19 ; whereas when , the graph is Type 2. The present work established that the direct product is Type 1 when , and yet is Type 1 when is multiple of 3 and Type 2 otherwise; whereas when , the graph is Type 2.
We contribute to Question 1 by giving positive evidences. A regular graph is conformable if admits a vertex coloring with colors such that the number of vertices in each color class has the same parity as chetwyndhiltoncheng . It is known that every Type 1 graph must satisfy the conformable condition. The converse is not true, but being conformable helps to identify whether a graph has the potential to be Type 1 or to be sure that it cannot be Type 1. In Theorem 2, we show a sufficient condition on the graph for the direct product of regular graphs to be conformable.
Let and be two regular graphs. If is conformable, then is conformable.
Since is by hypothesis conformable, let us consider a vertex coloring such that, for every , the color class has cardinality of the same parity as . Consider one of the projections that define the direct product . Therefore, we have a function , which is a vertex coloring of such that every color class consists of the vertices in the Cartesian product of the sets of vertices of and , denoted by , for .
We consider two cases. First, consider the case when is a graph of even order. In this case, or is even. Recall that . Consider a vertex coloring , defined by . Note that this function is actually obtained from by changing the codomain, thus this is also a vertex coloring of , but possibly there are empty color classes. An empty set has cardinality zero, which is of even parity. We have to prove that is conformable, that is, we have to prove that every non-empty color class of has an even cardinality. But, a non-empty color class of is of the form , where a color class of . Since has the same cardinality of and , if or is even, then is also even, and thus, is a conformable coloring of .
Now, consider the case when is a graph of odd order, that is and are odd. Since and are regular graphs, we have that and are even. Observe that all color classes of have odd cardinality. Next, we define a conformable coloring . Observe that all color classes of have odd cardinality, hence they must have at least one vertex. The idea is to remove even amount of vertices of each color class of . In this way, the parity of these color classes is preserved and additionally we define new color classes of cardinality one, for each of this removed vertices. Consider and consider, for each , a fixed element of . We define:
In addition, define . We construct a vertex coloring of by defining its color classes, each one is a subset of a color class of . Each color class of is , for , and , for . In order to show that is conformable, we have to show that has color classes and that each color class has an odd cardinality, as follows.
First, the number of color classes of is . Since
we have that = .
Finally, we prove that each color class of has an odd cardinality. Recall that is of odd order and has even degree. Since is a conformable coloring of and is of odd order, each is of odd cardinality, for . For each , has odd cardinality . Similarly, for each , has odd cardinality . Clearly, and , for have odd cardinality. Therefore, is a conformable coloring of .
By Theorem 2, we contribute to Question 1 since given regular graphs and with of Type 1, we know that is conformable. Conformable graphs of odd order and sufficient large maximum degree are Type 1, see Chew chew . For example, in Carol2021 , we use this fact together with Hamiltonian decompositions, to give a full classification of the total chromatic number of the direct product of complete graphs .
Next lemma presents a sufficient condition on the graph for the direct product to be Type 1, which leads to a corollary that answers Question 1 positively when one factor is Type 1 and the other is bipartite.
If is Type 1, then is Type 1.
Let be a total coloring of a Type 1 graph . Consider one of the projections that define the direct product and the composite function . Observe that .
First note that, as before, , when restricted to the vertices of , is a vertex coloring. Second, let and suppose this edge and its endvertex have the same color, that is . Thus, , a contradiction since is an endvertex of the edge in . Finally, let and be two adjacent edges of and suppose that these edges have the same colors, that is . Thus, , a contradiction since and are adjacent edges in . ∎
If is Type 1 and is bipartite, then is Type 1.
We remark that is Type 1, which agrees with Corollary 1, as is Type 1 and is a bipartite graph. We remark that is Type 1 if and only if is multiple of 3. The converse of Corollary 1 is not true, since there are many examples of a Type 2 graph such that the direct product of is Type . For instance, for even is Type 2, and yet is the complete bipartite graph minus a perfect matching, known to be Type 1 for yapbook .
Also, for two complete bipartite graphs and , the direct product is Type 2 if and only if and . Otherwise, it is Type 1. Indeed, note that if or , then is Type 1, by Corollary 1 since it is known that is Type 1 yapbook . On the other hand, if and , the graph is isomorphic to two copies of jha . It is known that is Type 2 yapbook and thus also .
If is bipartite and Type 1, then Corollary 1 implies that is Type 1 as well. In this context, we conclude by proposing the property for a general Type 1 graph:
Given a Type 1 graph , is the direct product Type 1 as well?
This work is partially supported by the Brazilian agencies CNPq (Grant numbers: 302823/2016-6, 407635/2018-1 and 313797/2020-0) and FAPERJ (Grant numbers: CNE E-26/202.793/2017 and ARC E-26/010.002674/2019).
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