A generalization of Kneser's conjecture

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 @g"""m(G). In view of Tucker-Ky Fan's lemma, we show that @g"""m(KG(n,k))[email protected](KG(n,k))-2=2n-4k+2 provided that [email protected]?83k. 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 @g"""m(KG(n,k))>=max{[email protected](KG(n,k))-10,@g(KG(n,k))}, where n>=2k>=4. Also, for n>=2k>=4, we conjecture that @g"""m(KG(n,k))[email protected](KG(n,k))-2.

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