On the chromatic number of regular graphs of matrix algebras

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