On graphs whose cycle space is spanned by their Hamilton cycles

Abstract

The cycle space of a graph $G$, denoted $\mathcal{C}(G)$, is a vector space over ${\mathbb F}_2$, spanned by all incidence vectors of edge-sets of cycles of $G$. If $G$ has $n$ vertices, then $\mathcal{C}_n(G)$ denotes the subspace of $\mathcal{C}(G)$, spanned by the incidence vectors of Hamilton cycles of $G$. We consider several known sufficient conditions for Hamiltonicity and show that an appropriate and fairly mild strengthening of each such condition in fact ensures the stronger property $\mathcal{C}_n(G) = \mathcal{C}(G)$. In particular, we consider the classical Chvátal-Erdős criterion and prove that (under various additional restrictions) if $n$ is odd and $κ(G) \geq c α(G)$, where $c$ is a sufficiently large absolute constant, then $\mathcal{C}_n(G) = \mathcal{C}(G)$. Moreover, considering the McDiarmid-Yolov criterion we prove that if $n$ is odd and $δ(G) \geq \max \left\{2 \tildeα(G) + 9, \tildeα(G) + 18 \right\}$, where $\tildeα(G)$ is the so-called bipartite independence number of $G$, then $\mathcal{C}_n(G) = \mathcal{C}(G)$. We also prove that if $n$ is odd and $G$ admits $16 α(G) + 12$ pairwise disjoint connected dominating sets, $\mathcal{C}_n(G) = \mathcal{C}(G)$. Finally, we consider an effective Chvátal-Erdős type criterion for bipartite graphs and prove that if $G$ is a balanced bipartite graph on $2n$ vertices, satisfying $α_{\emph{BIP}}(G) \leq 2 δ(G) - 24$, then $\mathcal{C}_{2n}(G) = \mathcal{C}(G)$.