On total and edge coloring some Kneser graphs

Abstract

In this work, we investigate the total and edge colorings of the Kneser graphs K(n, s). We prove that the sparse case of Kneser graphs, the odd graphs $O_k=K(2k-1,k-1)$, have total chromatic number equal to $\Delta (O_k) + 1$. We prove that Kneser graphs K(n, 2) verify the Total Coloring Conjecture when n is even, or when n is odd not divisible by 3. For the remaining cases when n is odd and divisible by 3, we obtain a total coloring of K(n, 2) with $\Delta (K(n,2)) + 3$ colors when $n \equiv 3~\hbox {mod}~4$, and with $\Delta (K(n,2)) + 4$ colors when $n \equiv 1~\hbox {mod}~4$. Furthermore, we present an infinite family of Kneser graphs K(n, 2) that have chromatic index equal to $\Delta (K(n,2))$.