Extremal regular graphs of given chromatic number

We define an extremal $(r|\chi)$-graph as an $r$-regular graph with chromatic number $\chi$ of minimum order. We show that the Tur{\' a}n graphs $T_{ak,k}$, the antihole graphs and the graphs $K_k\times K_2$ are extremal in this sense. We also study extremal Cayley $(r|\chi)$-graphs and we exhibit several $(r|\chi)$-graph constructions arising from Tur{\' a}n graphs.


Introduction
An r-regular graph is a simple finite graph such that each of its vertices has degree r.Regular graphs are one of the most studied classes of graphs; especially those with symmetries such as Cayley graphs.Let Γ be a finite group and let X = {x 1 , x 2 , . . ., x t } a generating set for Γ such that X = X −1 with 1 Γ ∈ X; a Cayley graph Cay(Γ, X) has vertex set consisting of the elements of Γ and two vertices g and h are adjacent if gx i = h for some 1 ≤ i ≤ t.Cayley graphs are regular but there exist non-Cayley vertex-transitive graphs.The Petersen graph is a classic example of this fact.
The girth of a graph is the size of its shortest cycle.An (r, g)-graph is an r-regular graph of girth g.An (r, g)-cage is an (r, g)-graph of smallest possible order.The diameter of a graph is the largest length between shortest paths of any two vertices.An (r; D)-graph is an r-regular graph of diameter D.
While the cage problem asks for the constructions of cages, the degreediameter problem asks for the construction of (r; D)-graphs of maximum order.Both of them are open and active problems (see [3,6]) in which, frequently, it is considered the restriction to Cayley graphs, see [4,5].
In this paper, we study a similar problem using a well-known parameter of coloration instead of girth or diameter.A k-coloring of a graph G is a partition Suppose that G is a (r|1)-graph.Hence G is the empty graph, then r = 0. Therefore, the extremal graph is the trivial graph.We can assume that 2 ≤ χ ≤ r + 1.
Next, we prove that for any r and χ such that 2 ≤ χ ≤ r + 1, there exists a Cayley (r|χ)-graph G.
We recall that the (n, k)-Turán graph T n,k is the complete k-partite graph on n vertices whose partite sets are as nearly equal in cardinality as possible, i.e., it is formed by partitioning a set of n = ak + b vertices (with 0 The (n, k)-Turán graph has chromatic number k, and size (see [1]) Lemma 2.1.The (ak, k)-Turán graph T ak,k is a Cayley graph.
Then, the graph Cay(Γ, X) is isomorphic to T ak,k .
Before to continue, we recall some definitions.Given two graphs H 1 and H 2 , the cartesian product H 1 H 2 is defined as the graph with vertex set V (H 1 ) × V (H 2 ) and two vertices (u, u ′ ) and (v, v ′ ) are adjacent if either u = v and u ′ is adjacent with v ′ in H 2 , or u ′ = v ′ and u is adjacent with v in H 1 .The following proposition appears in [10].
Proposition 2.2.The cartesian product of two Cayley graphs is a Cayley graph.
On the other hand, the chromatic number of H 1 H 2 is the maximum between χ(H 1 ) and χ(H 2 ), see [2].Now we can prove the following theorem.
Additionally, consider the graph H 2 = T b+1,b+1 = K b+1 .The graph H 2 has chromatic number b + 1 < χ and it is a b-regular graph of order b + 1.
To improve the lower bound we consider the (n, χ)-Turán graph T n,χ .Suppose G is an (r|χ)-graph.Let ς be a χ-coloring of G resulting in the partition (V 1 , V 2 , . . ., V χ ) with |V i | = a i for 1 ≤ i ≤ χ.Then the largest possible size of G occurs when G is a complete χ-partite graph with partite sets (V 1 , V 2 , . . ., V χ ) and the cardinalities of these partite sets are as equal as possible.This implies that nr 2 since G has size rn/2.After some calculations we get that An (r|χ)-graph G of n(r|χ) vertices is called extremal (r|χ)-graph.Similarly, a Cayley (r|χ)-graph G of c(r|χ) vertices is called extremal Cayley (r|χ)-graph.When χ − 1|r the lower bound and the upper bound of Theorem 2.4 are equal.We have the following corollary.
In the remainder of this paper we exclusively work with b = 0, that is, when χ − 1 is not a divisor of r.

Antihole graphs
A hole graph is a cycle of length at least four.An antihole graph is the complement G c of a hole graph G.Note that a hole graph and its antihole graph are both connected if and only if their orders are at least five.In this subsection we prove that antihole graphs of order n are extremal (r|χ)-graphs for any n at least six.There are two cases depending of the number of vertices.
Therefore we have the following result: Therefore, we have the following theorem.
A hole graph is also considered a 2-factor since is a spanning 2-regular graph.For short, we denote the disjoint union of j cycles of lenght i as jC i .
Let G be an union of cycles for a i ≥ 0 with i ∈ {3, 4, . . ., 2t}.Note that the complement G c of G is the join of the complement of cycles.
Theorem 2.7.The graph Moreover, we have the following results.

The case of r = χ
In this subsection, we discuss the case of r = χ = k, i.e., the (k|k)-graphs of minimum order.We have the following bounds so far: We prove that the upper bound is correct except for k = 4 and maybe for k = 6, 8, 10, 12.To achieve it, we assume that there exist (k|k)-graphs of order Now, we use a bound for the chromatic number arising from the Reed's Conjecture, see [9].We recall the clique number ω(G) of a graph G is the largest k for which G has a complete subgraph of order k.
Conjecture 2.10.For every graph G, It is known that the conjecture is true for graphs satisfying Equation 1, see [7].
Let H 1 be a clique of G and H 2 = G \ V (H 1 ).There is a set of k edges from V (H 1 ) and V (H 2 ).Therefore, if t = n − k ≤ k − 2 is the order of H 2 and m = (kt − k)/2 is the number of edges in H 2 , then m ≤ t 2 .
We obtain that k ≤ t, a contradiction.
Let H 1 be a clique of G and We obtain that k ≤ t + 1, hence, k = t + 1 and n has to be 2k − 2. Since every vertex v in V (H 2 ) has degree k in G, v has at least two neighbours in H 1 .By symmetry, G is the union of two complete graphs K k−1 with the addition of two perfect matchings between them.Its complement is a (k − 3)-regular bipartite graph.Any perfect matching of G c induce a (k − 1)-coloring in G, a contradiction.
We have the following results.
Lemma 2.11.For any k ≥ 3, If k is odd then the order of any k-regular graph is even, therefore: Corollary 2.12.For any k ≥ 3 an odd number, n(k|k) = c(k|k) = 2k.
We have that C c 7 is the extremal (4|4)-graph.Next, assume that k ≥ 6 is an even number and there exists a (k|k)-graph G of n = 2k − 1 vertices.Owing to the fact that χ(G) ≤ n − α(G) + 1 where α(G) is the independence number of G, we get that α(G) ≤ k.
In [7] was proved that the Reed's conjecture holds for graphs of order n satisfying χ > n+3−α 2 .In the case of the graph G, we have that It follows that ω(G) ≤ k ≤ ω(G) + 1. Newly, we have two cases: As we saw before, let H 1 be a clique of G and We obtain that k ≤ t, a contradiction.
Finally, we have the following theorem.

Non-Cayley constructions
In this section we improve the upper bound of n(r|χ) given on Theorem 2.4 by exhibiting a construction of graphs not necessarily Cayley.We assume that r is not a multiple of χ − 1, therefore 2 ≤ χ ≤ r.Additionally, we show two more constructions which are tight for some values.

Upper bound
To begin with, take the Turán graph T n,χ , for n = aχ + b, 0 < b < χ with r = a(χ − 1) + b and the partition ( Next, we define the graph G n,χ as the graph formed by two copies G 1 and G 2 of T n,χ with the addition of a matching between the vertices of degree r − 1 of G 1 and the vertices of degree r − 1 of G 2 in the natural way.In consequence, the graph G n,χ is an r-regular graph of order 2n and chromatic number χ.To obtain its chromatic number, suppose that T n,χ has the vertex partition V i , then the vertices of V i have the color i in G 1 and the vertices of where χ − 1|r − b with 0 ≤ b < χ.

The graph T * n,χ
In this subsection we give a better construction for some values of r and χ.
Consider the (aχ + b, χ)-Turán graph T aχ+b,χ such that χ > b ≥ 0 and partition We claim that a is even or χ − b is even.To prove it, assume that a and χ − b are odd.Hence, if b is even, then χ is odd, n = aχ + b is odd and r is odd, a contradiction.If b is odd, then χ is even, n = aχ + b is odd and r is odd, newly, a contradiction.Now, we define the graph T * n,χ of regularity r = a(χ − 1) + b − 1 as follows: If χ − b is even, the removal of a perfect matching between X i and X i+1 for all i ∈ {1, 3, . . ., χ − b − 1} of T n,χ produces T * n,χ .If χ − b ≥ 3 is odd then a is even, therefore, the removal of a perfect matching between X i and X i+1 for all i ∈ {4, 6 . . ., χ − b − 1} and a perfect matching between V ′ 1 and V ′′ 2 , V ′ 2 and V ′′ 3 , and V ′ 3 and V ′′ 1 where The graphs T * n,χ improve the upper bound given in Theorem 3.1 for some numbers n and χ: Hence, if χ−b χ−1 < 1, the construction gives extremal graphs, that is, when 1 < b.

3.3
The graph G a,c,t Consider the (at, t)-Turán graph T at,t with partition (V 1 , . . ., V t ).Now, we define the graph G a,c,t with 1 ≤ c < a as follows: consider two parts of (V 1 , . . ., V t ), e.g.V 1 and V 2 , and c vertices of these two parts {u 1 , . . ., The removal of the edges u i v j for i, j ∈ {1, . . ., c} when i = j (all the edges between {u 1 , . . ., u c } and {v 1 , . . ., v c } except for a matching) and the addition of the edges u i u j and v i v j for i, j ∈ {1, . . ., c} when i = j (all the edges between the vertices u i and all the edges between the vertices v i ) results in the graph G a,c,t .
The graph G a,c,t is a a(t−1)-regular graph of order at.Its chromatic number is t + c − 1 because the partition is a proper coloring with t + c − 1 colors.Moreover, the graph G a,c,t has a clique of t + c − 1 vertices, namely, the vertices {u 1 , . . ., u c , x 2 , . . ., x t } where x i ∈ V i for i ∈ {3, . . ., t} and The graphs G a,c,t improve the upper bound given in Theorem 2.4: Hence, if a c−1 t+c−2 < 1, the construction gives extremal graphs, that is, when

Small values
In this section we exhibit extremal (r|χ)-graphs of small orders.These exclude the extremal graphs given before.Table 1 shows the extremal (r|χ)-graphs for 2 ≤ r ≤ 10 and 2 ≤ χ ≤ 6.

Extremal (5|3)-graph
Suppose that G is an extremal (5|3)-graph of order 8, i.e., its order equals the lower bound given in Theorem 2.  Let G be an extremal (7|3)-graph.Its order is at least 11.Since its degree is odd, its order is at least 12.The graph T * 12,3 is an extremal (7|3)-graph.Now, suppose that G is an extremal (7|6)-graph.G has at least 9 vertices.Newly, because it has an odd regularity, G has at least 10 vertices.If this is the case, its complement is a 2 regular graph.The graph (2C 5 ) c has chromatic number 6.It is unique and it is Cayley.

Extremal (9|3)-graph
Any (9|3)-graph has 14 vertices, i.e., its order equals the lower bound given in Theorem 2.4.Suppose that there exist at least one of degree 14.Let (V 1 , V 2 , V 3 ) a partition by independent sets.Some of the parts, V 1 , has at least five vertices.Since the graph is 9-regular, V 1 has exactly 5 vertices.The induced graph of V 2 and V 3 is a bipartite regular graph of an odd number of vertices, a contradiction.Then, any (9|3)-graph has at least 16 vertices.

1 .
G = C c n for n = 2k and k ≥ 3. The graph G has regularity r = 2k − 3 and chromatic number χ = k.Any (2k − 3|k)-graph has an even number of vertices and at least rχ χ−1 =

4 .
Then its complement is 2 regular.That is, G c is C 8 or C 5 ∪ C 3 or C 4 ∪ C 4 .By Theorem 2.8, the complement of C 8 or C 5 ∪ C 3 or C 4 ∪ C 4 has chromatic number 4. Since G is 5-regular, a (5|3)-graph of order 9 does not exist and therefore 10 is the best possible.The graph G 5,2,2 is an extremal (5|3)-graph with 10 vertices.