On prisms, Möbius ladders and the cycle space of dense graphs

Abstract

Abstract. For a graph X, let f 0 pXq denote its number of vertices, δpXq its minimum degree and Z 1 pX; Z{2q its cycle space in the standard graph-theoretical sense (i.e. 1-dimensional cycle group in the sense of simplicial homology theory with Z{2-coefficients). Call a graph Hamiltongenerated if and only if the set of all Hamilton circuits is a Z{2-generating system for Z 1 pX; Z{2q. The main purpose of this paper is to prove the following: for every γ ą 0 there exists n 0 P Z such that for every graph X with f 0 pXq ě n 0 vertices, (1) if δpXq ě p 1 2`γ qf 0 pXq and f 0 pXq is odd, then X is Hamilton-generated, (2) if δpXq ě p 1 2`γ qf 0 pXq and f 0 pXq is even, then the set of all Hamilton circuits of X generates a codimension-one subspace of Z 1 pX; Z{2q, and the set of all circuits of X having length either f 0 pXq´1 or f 0 pXq generates all of Z 1 pX; Z{2q, (3) if δpXq ě p 1 4`γ qf 0 pXq and X is square bipartite, then X is Hamilton-generated. All these degree-conditions are essentially best-possible. The implications in (1) and (2) give an asymptotic affirmative answer to a special case of an open conjecture which according to [European J. Combin. 4 (1983), no. 3, p. 246] originates with A. Bondy.

Compatible Hamilton cycles in Dirac graphs

“…For example, one can try to show that a Dirac graph has many Hamilton cycles or that Maker can win a Hamiltonicity game played on edges of a Dirac graph. These extensions and other similar questions have been answered in [7] for the number of Hamilton cycles, in [6] and [9] for the number of edge-disjoint Hamilton cycles, in [10] for the cycle space generated by Hamilton cycles, and in [13] for Hamiltonicity of random subgraphs and for the Maker-Breaker games on Dirac graphs. Also, very recently, a number of related important problems on regular Dirac graphs, such as the existence of decomposition of its edge set into Hamilton cycles, have been settled in a series of papers starting from [16], using a structural result proved in [15].…”