Lower bounds for independence numbers of some locally sparse graphs

An m-distinct-coloring is a proper vertex-coloring c of a graph G if for each vertex v ∈ V , any color appears in at most one of N0(v), N1(v), . . ., and Nm(v), where Ni (v) is the set of vertices at distance i from v. In this note, we show that if G is C2m+1-free which is assigned an (m + 1)-distinct-coloring c, then α(G)c(G)1/m ≥ ( ∑ v c(v) 1/m ) , where c(G) is the number of colors used in c and c(v) is the number of different colors appearing in N1(v).Moreover,we obtain that ifG has N vertices and it contains neitherC2m+1 norC2m , thenα(G) ≥ ( (N log N )m/(m+1) ) . The algorithm in the proof for the first result is random, and that for the second is constructive.