A generalization of Kneser’s conjecture

Abstract

We investigate some coloring properties of Kneser graphs. A semi-matching coloring is a proper coloring c : V ( G ) → N such that for any two consecutive colors, the edges joining the colors form a matching. The minimum positive integer t for which there exists a semi-matching coloring c : V ( G ) → { 1 , 2 , … , t } is called the semi-matching chromatic number of G and denoted by χ m ( G ) . In view of Tucker–Ky Fan’s lemma, we show that χ m ( KG ( n , k ) ) = 2 χ ( KG ( n , k ) ) − 2 = 2 n − 4 k + 2 provided that n ≤ 8 3 k . This gives a partial answer to a conjecture of Omoomi and Pourmiri [Local coloring of Kneser graphs, Discrete Mathematics, 308 (24): 5922–5927, (2008)]. Moreover, for any Kneser graph KG ( n , k ) , we show that χ m ( KG ( n , k ) ) ≥ max { 2 χ ( KG ( n , k ) ) − 10 , χ ( KG ( n , k ) ) } , where n ≥ 2 k ≥ 4 . Also, for n ≥ 2 k ≥ 4 , we conjecture that χ m ( KG ( n , k ) ) = 2 χ ( KG ( n , k ) ) − 2 .