Generalized degree conditions for graphs with bounded independence number

Abstract

We consider a generalized degree condition based on the cardinality of the neighborhood union of arbitrary sets of r vertices. We show that a Dirac-type bound on this degree in conjunction with a bound on the independence number of a graph is sufficient to imply certain hamiltonian properties in graphs. For K,,,,-free graphs we obtain generalizations of known results. In particular we show:Theorem. Let r 2 1 and rn 2 3 be integers. Then for each nonnegative function f(r, rn) there exists a constant C = C(r, M, f(r, rn)) such that if G is a graph of order n ( n 2 r, n > rn) with S,(G) z-(a) G is traceable if 6(G) 2 r and G is connected; (b) G is harniltonian if 6(G) 1 r + 1 and G is 2-connected; (c) Gis hamiltonian-connected if 6(G) 2 r + ' 2 and Gis 3-connected.

Advances on the Hamiltonian Problem - A Survey

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