New Results on k-Independence of Graphs
Abstract
Let G=(V,E) be a graph and k≥0 an integer. A k-independent set S⊆G is a set of vertices such that the maximum degree in the graph induced by S is at most k. Denote by αk(G) the maximum cardinality of a k-independent set of G. For a graph G on n vertices and average degree d, Turán's theorem asserts that α0(G)≥nd+1, where the equality holds if and only if G is a union of cliques of equal size. For general k we prove that αk(G)≥(k+1)nd+k+1, improving on the previous best bound αk(G)≥(k+1)n⌈d⌉+k+1 of Caro and Hansberg [E-JC, 2013]. For 1-independence we prove that equality holds if and only if G is either an independent set or a union of almost-cliques of equal size (an almost-clique is a clique on an even number of vertices minus a 1-factor). For 2-independence, we prove that equality holds if and only if G is an independent set. Furthermore when d>0 is an integer divisible by 3 we prove that α2(G)≥3nd+3(1+125d2+25d+18).