Vizing-Goldberg type bounds for the equitable chromatic number of block graphs

An equitable coloring of a graph G is a proper vertex coloring of G such that the sizes of any two color classes differ by at most one. In the paper, we pose a conjecture that offers a gap-one bound for the smallest number of colors needed to equitably color every block graph. In other words, the difference between the upper and the lower bounds of our conjecture is at most one. Thus, in some sense, the situation is similar to that of chromatic index, where we have the classical theorem of Vizing and the Goldberg conjecture for multigraphs. The results obtained in the paper support our conjecture. More precisely, we verify it in the class of well-covered block graphs, which are block graphs in which each vertex belongs to a maximum independent set. We also show that the conjecture is true for block graphs, which contain a vertex that does not lie in an independent set of size larger than two. Finally, we verify the conjecture for some symmetric-like block graphs. In order to derive our results we obtain structural characterizations of block graphs from these classes.

READ FULL TEXT VIEW PDF

Authors

page 1

page 2

page 3

page 4

06/13/2020

Total Coloring for some classes of Circulant graphs

The Total coloring conjecture states that any simple graph G with maximu...
06/13/2020

Total Coloring for some classes of Cayley graphs

The Total coloring conjecture states that any simple graph G with maximu...
09/27/2020

Graph theoretic and algorithmic aspect of the equitable coloring problem in block graphs

An equitable coloring of a graph G=(V,E) is a (proper) vertex-coloring o...
08/14/2019

Equitable vertex arboricity of d-degenerate graphs

A minimization problem in graph theory so-called the equitable tree-colo...
01/14/2020

Strong coloring 2-regular graphs: Cycle restrictions and partial colorings

Let H be a graph with Δ(H) ≤ 2, and let G be obtained from H by gluing i...
02/11/2022

Perfect Matchings and Quantum Physics: Progress on Krenn's Conjecture

In 2017, Krenn reported that certain problems related to the perfect mat...
06/18/2020

An Alon-Boppana theorem for powered graphs and generalized Ramanujan graphs

The r-th power of a graph modifies a graph by connecting every vertex pa...
This week in AI

Get the week's most popular data science and artificial intelligence research sent straight to your inbox every Saturday.

1 Introduction

A hypergraph is a pair , where is an element set of vertices of and is a family of non-empty subsets of called edges or hyperedges. Let denote the set of integers . A -coloring of hyperedges of is a mapping such that no two edges that share a vertex get the same color (number). An edge -coloring of is equitable if each color class is of size or . In other words, an equitable edge coloring of may be seen as a partition of the hyperedge set into independent subsets such that , for each . The smallest such that admits an equitable edge -coloring is called the equitable chromatic index and is denoted by .

For a hypergraph we need to define the concept of its line graph/host graph (cf. Fig. 1). The line graph is a simple graph representing adjacencies between hyperedges in . More precisely, each hyperedge of is assigned a vertex in and two vertices in are adjacent if and only if their corresponding hyperedges share a vertex in . We say that a hypergraph has an underlying (host) graph (spanned on the same set of vertices) if each hyperedge of induces a connected subgraph in . Furthermore, it is assumed that for each edge in there exists a hyperedge in such that .

It is easy to notice that an edge coloring of a hypergraph is equivalent to a vertex coloring of its line graph. A -coloring of vertices of a simple graph is an assigning of colors from the set to vertices in such a way that no two adjacent vertices receive the same color. A vertex -coloring is equitable if each color class is of size or . The smallest such that admits an equitable vertex coloring is called the equitable chromatic number of and is denoted by . Moreover, note that for a general graph if it admits an equitable vertex -coloring it does not imply that it admits an equitable vertex -coloring (cf. for example and ). That is why we also consider the concept of equitable chromatic spectrum, i.e. the set of colors admitting equitable vertex coloring of the graph. The smallest such that admits an equitable vertex -coloring for every is called the equitable chromatic threshold and is denoted by . If then we say that the equitable chromatic spectrum of is gap-free.

(a)      (b)

Figure 1: (a) An example of hypertree (hyperedges depicted with solid lines) with an exemplary equitable edge 4-coloring (colors in brackets) and an exemplary host graph (dotted edges). (b) The corresponding line graph with the corresponding equitable vertex 4-coloring.

Despite the fact that the corresponding problem for simple graphs has been widely studied (for interesting surveys, see furm:en_book , lih ), its generalization to hypergraphs does not seem to have been addressed in the literature. To the best of our knowledge there is no paper in the literature that concerns the problem of equitable edge coloring of hypergraphs with the definition given above. Hypergraphs in general are very useful in real-life problems modeling, for example in chemistry, telecommunications, and many other fields of science and engineering hyper . They have also applications in image representation image . Thus, generalization of equitable coloring of simple graphs to hypergraphs seems to be justified. It is known that the model of equitable coloring of simple graphs has many applications, among others in task scheduling (see furm:4sch , obsz:jastrz ). Every time, when we have to divide a system with binary conflict relations into equal or almost equal conflict-free subsystems, we can model this situation by means of equitable graph coloring.

In the paper we study chordal graphs, their subclasses, and the complexity status of the problem of equitable coloring for them. A graph is chordal if every cycle of length at least 4 has a chord. It is known (cf. duchet ; mckee ) that a graph is chordal, if and only if it is a line graph of a hypertree, where hypertree is defined as a hypergraph that has an underlying tree. Thus equitable edge coloring of hypertrees is equivalent to equitable vertex coloring of chordal graphs (cf. Fig. 1). On the other hand, we know (cf. bounded , bod:part ) that the problem of equitable vertex coloring of interval graphs is NP-hard. Since each interval graph is also chordal, we have also NP-hardness of the problem for chordal graphs. In consequence:

Corollary 1.1.

The problem of an equitable edge-coloring for hypertrees is NP-hard.

Bodlaender bounded proved that the problem of equitable -coloring can be solved in polynomial time for graphs with given tree decomposition and for fixed . The treewidth of a chordal graph equals the maximum clique size minus one. Bodlaender bounded proved also that the problem of an equitable -coloring is solvable in polynomial time for graphs with bounded degree even if is a variable.

Corollary 1.2.

The problem of an equitable -coloring is solvable in polynomial time for chordal graphs with bounded maximum clique size.

On the other hand, Gomes et al. EqParamFPT proved that, when the treewidth is a parameter to the algorithm, the problem of equitable vertex coloring is W[1]-hard. Thus, it is unlikely that there exists a polynomial time algorithm independent of this parameter. In this paper, we address the problem in block graphs, which are the graphs with every 2-connected component being a clique. A clique of a graph is a maximal complete subgraph of . For block graphs, it is shown in EqParamFPT that the problem is W[1]-hard with respect to the treewidth, diameter and the number of colors. This in particular means that under the standard assumption FPTW[1] in parameterized complexity theory, the problem is not likely to be polynomial time solvable in block graphs.

In what follows when we refer to equitable coloring we mean equitable vertex coloring unless stated differently. For a graph let be the size of the largest independent set in , while is the size of the largest independent set that contains the vertex in . Define:

Clearly, , and if and only if every vertex of lies in a maximum independent set of . Such graphs are known in the literature as well-covered graphs plummer . A simplicial vertex is a vertex that appears in exactly one clique of a graph. A cut vertex in a connected graph is a vertex that is disconnected. In a block graph we define a clique as pendant if it contains exactly one cut-vertex, while a clique is internal if all its vertices are cut-vertices. Vertices of an internal clique are called internal vertices. For a graph let be the size of the largest clique of . For every graph, not necessarily a block graph, it is known that

(1)

Indeed, the equitable chromatic number of a graph cannot be less than its clique number. Moreover, it cannot be less than . The latter follows from the assumption that one color is used exactly times, and any other color can be used at most times. It turns out that the number of colors given by the expression on the right side of the inequality is not sufficient to color equitably every block graph. For example, take a clique of size , , and add pendant cliques of size to each vertex (cf. Fig. 2). It can be easily checked that , , , while . Observe that Thus,

One can easily check that there is no equitable -coloring of such graphs. On the other hand, it is easy to show that this graph is equitably -colorable, hence .

Figure 2: The case of the example.

The gap between and in the example above is one. This led us to the following conjecture, which is similar to the classical theorem of Vizing for graphs and the Goldberg conjecture for multigraphs:

Conjecture 1.3.

For any block graph , we have:

We have confirmed the conjecture for all block graphs on at most 19 vertices, using a computer. Moreover, the conjecture is true for forests, i.e. for block graphs with forests . Since the class of connected block graphs in which each cut vertex is on exactly two blocks is equivalent to line graphs of trees harary , we have for a tree and its line graph . As trees are of Class 1 and for a tree , then we have for connected block graphs in which each cut vertex is on exactly two blocks. Thus our conjecture is true for such block graphs. Moreover, since an arbitrary graph is equitably edge -colorable for every furm:en_book , then and the equitable chromatic spectrum of block graph in which each cut vertex is on exactly two blocks is gap-free. In this paper we prove the conjecture for well-covered block graphs, using the unusual tool of Ferrers matrix (Section 2). Moreover, we prove our Vizing-Goldberg type conjecture for block graphs with small value of (Section 4) as well for some symmetric-like block graphs (Section 3).

Finally, we would like to draw the reader attention +to the fact that the problem of block graphs coloring is closely related to very well known EFL conjecture formulated by Erdős, Faber, and Lovász in 1972 erdos . They supposed that if complete graphs, each having exactly vertices, have the property that every pair of complete graphs has at most one shared vertex, then the union of the graphs can be colored with colors. Note that some block graphs are among those graphs that are affected by the conjecture. The only condition that they must fulfill is the size of all blocks is the same, and is equal to . Note that the EFL conjecture implies that classical chromatic number of the graphs for which the conjecture holds is equal to their clique number. Hence, our results that say that some block graph can be equitably colored with colors partialy confirm EFL conjecture - cf. Theorems 2.5 and 3.3.

2 Well-covered block-graphs

In this section we confirm Conjecture 1.3 for well-covered block graphs, i.e. for graphs fulfilling the condition . In this case we have the following chain of inequalities

Hence the conjecture states that any such block graph must be equitably - or -colorable. In fact, we will show that well-covered block graphs are equitably -colorable for all .

2.1 Characterization

We start with a recursive characterization of well-covered block graphs. There is one basic class of such graphs, namely complete graphs. Now, we define the following operation: let be a block graph and let be a vertex of . Add a clique of size at least 2 to (), and add one pendant clique to each vertex of except . With this, all vertices of will become cut-vertices and, in consequence, becomes an internal clique. Let us note that the added pendant cliques may have different sizes. For further purpose we take the following notation. Let . We add a clique of size , , with pendant cliques to graph by identifying one vertex of with . Let . Let denote pendant clique added to vertex , ,

. We order the pendant cliques due to their sizes in a non-increasing way creating a vector

, (cf. Fig. 4). Every , , corresponds to the size of one pendant clique. We assume that and this term corresponds to the vertex .

Theorem 2.1.

Let be a well-covered block graph with a largest clique size . Then is either a basic graph, i.e. a clique of size at most , or can be obtained from a smaller well-covered block graph with by using the operation defined above, such that all involved cliques are of size at most .

Proof.

We prove this theorem in three steps. First of all, let us note that our basic class satisfies the condition . Indeed, since is a clique, then clearly .

Now, let us show that if we have a block graph with and we apply the operation given above (Case 1), then we will get a block graph with . Assume that is the vertex of to whom we have added a clique of size and have added pendant cliques to the vertices of except . Clearly, . In order to complete the proof of this step, let us show that any vertex of lies in an independent set of of size . Let be any vertex of . If lies in or is one of simplicial vertices of pendant cliques adjacent to , then the statement is easy. Let us assume that is an internal vertex of . If it is then again the statement is easy. If it is a vertex of different from , then let be a vertex of adjacent to . We can extend to an independent set of of size , because . Observe that does not belong to this independent set. Now add to it, and add simplicial vertices of pendant cliques (one simplicial vertex per pendant clique, except the one containing ). Observe that the resulting set is independent and its size is . Thus, .

In order to complete the proof of the theorem, let us show that if is a block graph with , largest clique size and outside from the basic class, then it can be obtained from a block graph with and the largest clique size by using our operation. First of all, let us observe that contains no two intersecting cliques, such that both of them contain simplicial vertices. If has two such cliques, then let be a vertex in their intersection. Extend to an independent set of of size . Now, replace with two simplicial vertices of these cliques. Clearly, the resulting set is independent and its size is , which is a contradiction.

Now let us remove all pendant cliques of except their cut-vertices. Because of the tree-structure of the block graph, we can always find a pendant clique in the resulting graph. Observe that if we put back the removed cliques, then each vertex of except the cut-vertex lies in a pendant clique of . Let this clique be of size . Observe that the vertices of different from are intersecting with pendant cliques, one pendant clique per vertex different from . Let be the graph obtained from by removing the pendant cliques intersecting and the vertices of different from . Observe that is a block graph with . Moreover, . We claim that . In order to see this, it suffices to show that any vertex of lies in an independent set of of size . Since , lies in an independent set of of size . Without loss of generality, we can assume that this independent set takes simplicial vertices from pendant cliques intersecting (one simplicial vertex per pendant clique). Observe that the number of these vertices is . Hence, if we remove these vertices, we will obtain an independent set of having size . Thus, and the proof is complete. ∎

It worths to mention that our characterization completes the knowledge on characterization of different well-covered classes of graphs given in the literature (cf.prisner ).

2.2 Some auxiliaries

In the proof of the main result of this section we use a slightly modified concept of Ferrers matrix. Let be any sequence with , . It can be visualized by matrix of zeros and ones called Ferrers matrix for and defined by a sum of its row vector and the properties: (1) , (2) if then for all . In our concept modified Ferrers matrix is also defined by row vector but, in addition, we have property , while the second condition is modified to: if then for all excluding (cf. Fig. (3a)).

a)                  b)
                 
Figure 3: a) Modified Ferrers matrix for and ; b) An example of matrix created due to the instructions given in the proof of Theorem 2.2 corresponding to the matrix with vector .

Let denote the vector of column sums in the matrix . Given two arbitrary sequences and , such that , we say that the vector is dominated by the vector if for all positive integers , where for all and for all . We denote it by .

Theorem 2.2.

Let and be two non-increasing sequences such that . Moreover, let be the modified Ferrers matrix for . If is dominated by then there exists matrix such that and , and for every .

Proof.

The proof is based on the proof of sufficiency of Gale-Ryser theorem gale . That proof is constructive. It starts from Ferrers matrix and requires finite number of applications of the following claim proved in gale .

Claim 2.3.

Given matrix of zeros and ones such that , and , we can find a matrix of zeros and ones such that , , and where is the ordinary Euclidean norm.

Matrix is constructed in the following way. First, we choose as minimal such that sum of -th column of is greater than and choose as minimal such that sum of -th column of is lower than . Next, we can find a row index (there are at least two choices) such that and . Matrix is build from by swapping elements and .
To prove Theorem 2.2 we start from modified Ferrers matrix and carefully apply Claim 2.3. In every step, when we choose row we take one such that is not of the form (cf. Fig. (3b)). ∎

Lemma 2.4.

If vector from Theorem 2.2 fulfills the following conditions: while for all , and , then is dominated by .

Proof.

Let . Since (due to the definition of modified Ferrers matrix), , and vector is semi-balanced, i.e. fulfills the assumption of this lemma, we are sure that . Thus, we have for all for some , . To the contrary, let us assume that is not dominated by , i.e. , , and hence . Since , there must exist an index such that . Since and vector is non-increasing, such situation is impossible. ∎

2.3 The main result

We are ready to prove

Theorem 2.5.

Let be a well-covered block graph. Then is equitably -colorable for all .

Proof.

We use the recursive characterization of well-covered block graphs. It is clear that the theorem holds for the basic class of well-covered block graphs, i.e. for cliques. We assume that we have an equitable -coloring of (coloring ) and we show that it is possible to extend it into the entire graph for every .

The graph is given by a graph and the vector that describes the sizes of added cliques (cf. the characterization given in Subsection 2.1). Since an extended -coloring of must be equitable, we can calculate how many times each color should be used in a coloring of , , taking into account the equitable -coloring of . Let be a non-increasing vector of cardinalities of color classes in a coloring of . Moreover, let be a vector of colors such that the cardinality of color is equal to in a desirable coloring of (cf. Fig. 4). Note that there is at most one term in the sequence (), such that there is a term fulfilling inequality . For the remaining terms we have , and .

Figure 4: An example of graph with a counter of an equitable 6-coloring of given by the sequence . Then and while the vector of the corresponding colors is, for example, as follows: .

On the further purpose we claim that the following property of the coloring of holds.

Claim 2.6.

Graph may be recolored in such a way that the color assigned to is and term in vector represents color , i.e. .

Proof.

All we need is to prove that after the recoloring of graph color 1 is one of the colors with a largest cardinality in the desirable coloring of .

Since the initial -coloring of graph is equitable then the difference between cardinalities of any two color classes is at most 1. Let us consider two cases: (1) the initial color of , , is a color of a largest cardinality in the coloring of . Then we do not have a term in the sequence such that the difference between this term and any other term from this sequence is 2. Moreover, the cardinality of color is represented by a term of the sequence with the highest value. Thus, we may recolor graph in such a way that will obtain 1 and represents color 1; (2) the initial color is a color of the smallest cardinality in the coloring of . In this case we have two possible situations:

  • we do not have a term in that exceeds any other term by exactly 2 - then the cardinality of color is represented by a term of the sequence with the highest value,

  • we have a term in that exceeds any other term by exactly 2, but it corresponds to color .

In both cases, after recoloring to 1, represents color 1. ∎

To achieve the desirable equitable -coloring of we start from the equitable -coloring of such that vertex is colored with 1. Next, we build the modified Ferrers matrix for vector . Observe that there is at most one term in the sequence (), such that there is a term fulfilling inequality , while for the remaining terms we have , . By Lemma 2.4, we may apply Theorem 2.2 - vector corresponds to vector while vector corresponds to vector . As a result, we get a matrix such that and (cf. Fig. 3b)). Now, the desirable coloring of can be read from the matrix in the following way. Let the pendant clique of size be represented by the term in sequence . Then, , and the remaining vertices in are colored with all colors such that and (cf. Fig. 5). ∎

Figure 5: An equitable coloring of read from matrix given in Figure 3b).
Corollary 2.7.

The equitable chromatic spectrum of well-covered block graphs is gap-free.

3 Symmetric-like block graphs

In this section, we present some results that deal with symmetric-like block graphs. A symmetric block graph is a block graph whose all blocks are cliques of size , each cut vertex belongs to exactly blocks, and the eccentricity of simplicial vertices is same. Inspired by this definition, we define some other subclasses of block graphs and confirm Conjecture 1.3 for them.

3.1 Class ,

We consider a subclass of block graphs that is defined recursively. Let be a clique of size . A graph , , is built from by adding to each simplicial vertex pendant cliques, each one of size (cf. Fig. 6 with ).

Figure 6: Graph with an exemplary independent set of the largest size depicted by black vertices.

Note that

It is easy to observe that the largest independent set in a graph is formed by simplicial vertices - one such a vertex from each pendant clique - and appropriate cut vertices from every second ”level” (cf. Fig. 6). Thus, we have

Lemma 3.1.

For every graph being a , , we have

Proof.

It is enough to observe that an independent set of the smallest size is designated by a cut vertex from any pendant clique. ∎

Lemma 3.2.

For every graph being a , , we have

Proof.

Observe that in the construction of block graph we always work with cliques of size , hence for such graphs, we have . Thus, in order to prove the above equality, it suffices to show that

We split the proof of this into three cases. The first case is when . In this case, we have a clique of size , and the inequality is trivially true. Now, we prove when is even. Hence . As we stated in the introduction, we can assume that and .

In this case, observe that

(2)

In order to see this, observe that for case , it follows from the fact that . On the other hand, if , then

hence

which clearly implies the inequality (2). Now, by elementary manipulations, it can be shown that inequality (2) implies

Since , we have

Again, simple manipulations imply that this inequality is the same as

The latter implies

which is the bound we were trying to prove. This just follows from the formula for the sum of members of a geometric progression and the above mentioned formulas for , and .

Finally, we are left with the case and is odd. This case can be done in a way similar to the case even. ∎

To confirm Conjecture 1.3 for graphs we prove the following

Theorem 3.3.

Every block graph , , has an equitable -coloring.

Proof.

We start from -coloring of . Then we extend the coloring for vertices from next levels, for , , , . The -coloring is determined (due to permutations of colors) and it is easy to show that the coloring is equitable, even strongly equitable. This means, each color is used exactly the same number of times. ∎

Corollary 3.4.

There is infinite family of block graphs for which we have arbitrarily large difference and such graphs demand colors to be equitably colored.

Remember that we have given an example of infinite family of block graphs defined by the condition: also demanding colors to be equitably colored, in Section 2.

In conclusion, we have

Corollary 3.5.

Conjecture 1.3 holds for graphs.

Taking into account our consideration on the equitable chromatic spectrum for well-covered block graphs in Section 2, we may state an open question for graphs.

Question 3.6.

Is it true that graph , , is equitably -colorable for every ? In other words: Is the equitable chromatic spectrum gap-free for graphs?

3.2 Class

In the case of symmetric block graphs and block graphs the number of cliques to which a cut-vertex belongs was determined. Now, we weaken this condition and define the subclass of block graphs, denoted by , as the subclass of block graphs such that every block is a clique of size and every cut vertex belongs to at most blocks. In this subsection, we confirm Conjecture 1.3 for graphs from the class . Observe that for . We show that each such graph admits an equitable -coloring. We need some auxiliary definitions.

Definition 3.7.

A graph (not necessarily block graph) with designated vertex is of type:

  • if is equitably -colorable with colors: , , , and that occur and times, respectively, and vertex is colored with ;

  • if is equitably -colorable with colors: , , , and that occur and times, respectively, and vertex is colored with .

Observe that every graph from the class includes a vertex of degree .

Lemma 3.8.

A connected graph , , with any vertex of degree is of type (T1) or (T2).

Proof.

We will prove this lemma by induction on the order of graph . First, note that (one isolated vertex ) is of type (T1) with , and the triangle, with any of its vertices, is of type (T2).

Now, consider a graph on vertices, , and assume that lemma is true for graphs from of order smaller than . Let be a vertex in of degree 2. It belongs only to one block and it has two neighbors, let us say and . The vertex (resp. ) belongs to at most 3 blocks, thus it may be included in two other triangles. Let us consider the connected subgraph of that contains . We name it by . Now, let us consider . It consists of at most two connected components: and . Let (resp. ) be (resp. ) after adding vertex and restoring the original edges between and vertices of (resp. . If consists of less than two components, then an appropriate graph or/and is a single vertex (cf. Fig. 7). Similarly, we define subgraphs and for vertex (cf. Fig. 7). Every graph is or it belongs to with , . In both cases, every graph is of type (T1) or (T2).

We have six possibilities, up to an isomorphism:

  • with vertex is of type (T1), with vertex is of type (T1), with vertex is of type (T1), with vertex is of type (T1). For colors in and we use the permutation . For colors in and we use . After we color vertex in color , the graph is of type (T2).

  • with vertex is of type (T1), with vertex is of type (T1), with vertex is of type (T1), with vertex is of type (T2). For colors in and we use the permutation . For colors in and we use . After we color vertex in color , the graph is of type (T1).

  • with vertex is of type (T1), with vertex is of type (T1), with vertex is of type (T2), with vertex is of type (T2). For colors in and in we use the permutation . For colors in we use , and in we use . After we color vertex in color , the graph is of type (T2).

  • with vertex is of type (T1), with vertex is of type (T2), with vertex is of type (T1), with vertex is of type (T2). For colors in we use the permutation and in we use the permutation . For colors in and we use . After we color vertex in color , the graph is of type (T2).

  • with vertex is of type (T1), with vertex is of type (T2), with vertex is of type (T2), with vertex is of type (T2). For colors in and we use the permutation . For colors in we use , and in we use . After we color vertex in color , the graph is of type (T1).

  • with vertex is of type (T2), with vertex is of type (T2), with vertex is of type (T2), with vertex is of type (T2). For colors in and we use the permutation . For colors in and we use . After we color vertex in color , the graph is of type (T2).

Figure 7: An example of graph

Corollary 3.9.

Conjecture 1.3 holds for every .

Note that this result cannot be extended to large values of . In other words, it is not true that each is equitably -colorable (cf. Fig. 8).

Figure 8: An example of a block graph from class demanding more than 5 colors to be equitably colored. Here

On Figure 9, we give an example of a block graph from that attains the upper bound in Conjecture 1.3. In this example, , and . Thus,

It can be easily seen that the only way to partition 11 vertices of this graph into three almost equal size is to have 4, 4 and 3 vertices in each set. Using this observation, it can be shown (with a reasoning similar to the graph from Figure 2) that this graph is not equitably 3-colorable. On the other hand, it is equitably 4-colorable (see the coloring on Figure 9).

Figure 9: A graph from the class attaining the upper bound in Conjecture 1.3.

4 Block-graphs with small

In this section, we prove our conjecture for block-graphs with small value of . We start with . Note that any graph with has one vertex adjacent to any other vertex of . In the case of block graphs, a graph consists of cliques , , sharing one common vertex. Let denote the size of the clique , . Assume that cliques are ordered according to their sizes, in a non-decreasing way, i.e. . We denote such a block graph by , . Let us think about equitable coloring of , , with the smallest number of colors. The color assigned to the universal vertex of cannot be used to color any other vertex in . Since coloring must be equitable, every other color in this coloring can be used at most twice. Thus, we need to partition the vertex set of into minimum number of color classes of size at most 2. This problem is equivalent to finding a maximum matching in the complement of . Note that is a complete multipartite graph . The maximum matching problem in complete multipartite graphs was considered in multi . Due to the results given in the paper, we have the following.

Theorem 4.1 (Thm. 1 in multi , original notations).

Let be a complete multipartite graph with vertices in the th part, labeled so that . If , then:

  1. the number of edges in any maximum matching is ;

  2. a maximum matching can be obtained by connecting all vertices in the parts with vertices to vertices in the part with vertices.

Considering our block graph , , such that fulfills the assumption of Theorem 4.1, which implies that in an optimal equitable coloring of we have , color classes of size 2 and color classes of size 1. Thus, we have color classes in total. Hence, in this case.

Theorem 4.2 (Thm. 3 in multi , original notation).

Given any complete multipartite graph, , where and , the number of edges in any maximum matching is .

Note, that considering our block graph , , such that fulfills the assumption of Theorem 4.2, which implies that in an optimal equitable coloring of we have color classes of size two and 1 or 2 (depending on parity of ) color classes of size one. Thus, we have color classes in total. Hence, .

Corollary 4.3.

Conjecture 1.3 is true for block graphs with .

Next, we give a characterization of connected block graphs that satisfy the condition . First, we observe some properties of such graphs.

Observation 4.4.

For each with there exists a maximal independent set such that at least one out of and is a cut vertex.

Proof.

If is a cut vertex, we are done. So let us assume that is a simplicial vertex that belongs to a clique . Since , the second vertex, let it be , from including , must belong to another clique, . Moreover, . Otherwise the maximal independent set could be replaced by the vertex from , a contradiction with . If is a cut vertex, we are done. So, let us assume that and are simplicial vertices. But then one of them, let us say , can be replaced in by the cut vertex belonging to and lying on the path , what finishes the proof. ∎

Observation 4.5.

Let be a maximal independent set in , , , and let be a cut vertex. Then only one of the following conditions hold: (1) There is exactly one clique, let us say , on the path , different from and . (2) There are exactly two cliques on the path , different from and , but the clique intersecting non empty clique is .

Any other situation according the cliques on the path is impossible.

Proof.

It is easy to observe that there must exist at least one clique on the path different from and . We have already given the reason in the proof of Observation 4.4. Let us consider the situation when we have exactly one clique, different from and , on the path . If the clique were of size greater than two, we could add any of its simplicial vertices to the independent set - contradiction with is maximal. If we had two or more cliques on the path , we also would increase the size of