A note on dominating cycles in 2-connected graphs

Let G be a 2-connected graph on n vertices such that d(x) + d(y) + d(z) n for all triples of independent vertices x, y, z. We prove that every longest cycle in G is a dominating cycle unless G is a spanning subgraph of a graph belonging to one of four easily specified classes of graphs

Cubic graphs with large circumference deficit

The circumference $c(G)$ of a graph $G$ is the length of a longest cycle. By exploiting our recent results on resistance of snarks, we construct infinite classes of cyclically $4$-, $5$- and $6$-edge-connected cubic graphs with circumference ratio $c(G)/|V(G)|$ bounded from above by $0.876$, $0.960$ and $0.990$, respectively. In contrast, the dominating cycle conjecture implies that the circumference ratio of a cyclically $4$-edge-connected cubic graph is at least $0.75$. In addition, we construct snarks with large girth and large circumference deficit, solving Problem 1 proposed in [J. H\"agglund and K. Markstr\"om, On stable cycles and cycle double covers of graphs with large circumference, Disc. Math. 312 (2012), 2540--2544]

Cycles containing many vertices of large degree

AbstractLet G be a 2-connected graph of order n, r a real number and Vr=v ϵ V(G)¦d(v)⩾r. It is shown that G contains a cycle missing at most max {0, n − 2r} vertices of Vr, yielding a common generalization of a result of Dirac and one of Shi Ronghua. A stronger conclusion holds if r⩾13(n+2): G contains a cycle C such that either V(C)⊇Vr or ¦ V(C)¦⩾2r. This theorem extends a result of Häggkvist and Jackson and is proved by first showing that if r⩾13(n+2), then G contains a cycle C for which ¦Vr∩V(C)¦is maximal such that N(x)⊆V(C) whenever x ϵ Vr − V(C) (∗). A result closely related to (∗) is stated and a result of Nash-Williams is extended using (∗)