On the chromatic number of regular graphs of matrix algebras

Abstract

Let R be a ring and Z(R) be the set of zero divisors of R. The regular graph of R, denoted by Γ(R) is the graph with vertex set R\Z(R) and {X, Y } is an edge if X +Y ∈ Z(R). We prove that the chromatic number of Γ(M n (F q )) is at least (q/4) n/2 , where M n (F q ) is the ring of n × n matrices over F q , q being an odd prime power. This proves that the chromatic number of Γ(M n (F alg p )) is infinite, answering a case of a question posed in BCC22.

On the chromatic number of structured Cayley graphs

“…More precisely, we will show that as q → ∞, we have χ(G G,S (F q )) → ∞. In fact, similar to [26] we will establish a polynomial lower bound for χ(G G,S (F q )). Further, we will also show that the large chromatic number is not due to the existence of large cliques; in fact, as we will see, the clique number of G G,S (F q ) is uniformly bounded as q → ∞ (see Proposition 1.7).…”

“…Using Lang-Weil bound and representation theory of finite simple groups of Lie type, we will establish lower bounds on the chromatic number of a large family of these graphs. As a corollary we obtain a lower bound for the chromatic number of certain Cayley graphs associated to the ring of n × n matrices over finite fields, establishing a result for the case of SLn parallel to a theorem of Tomon [26] for GLn. Moreover, using Weil's bound for Kloosterman sums we will also prove an analogous result for SL2 over certain finite rings.2010 Mathematics Subject Classification.…”

“…Interestingly enough, the answer to this question is negative. In fact, Tomon [26] proved that the chromatic number of the finite graph G n (F q ) is at least (q/4) n/2 , under the assumption that q = p f is odd, which immediately implies that G n (F p ) is infinite. Here F p denotes an algebraic closure of F p , and can be naturally viewed as the union of all finite fields F q for q = p f .…”

The sum of nonsingular matrices is often nonsingular

“…We will refine their result giving almost sharp bound on the clique number. Tomon [11] showed that the chromatic number of this graph is at least (q/4) ⌊n/2⌋ .…”

New Algorithm for Chromatic Number of Graphs and their Applications

“…As has been mentioned in [2] a greedy method would be to iteratively pick, in a graph , an uncolored vertex , and to color it with the smallest color which is not yet used by its neighbors . Such a coloring will obviously stay proper until the whole vertex set is colored, and it never uses more than ∆( ) + 1 different colors, where ∆( ) is the maximal degree of , as in the procedure no vertex will ever exclude more than ∆( ) colors (for more details see [3][4][5][6][7][8]). Consider is a graph.…”