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]

Box representations of embedded graphs

A $d$-box is the cartesian product of $d$ intervals of $\mathbb{R}$ and a $d$-box representation of a graph $G$ is a representation of $G$ as the intersection graph of a set of $d$-boxes in $\mathbb{R}^d$. It was proved by Thomassen in 1986 that every planar graph has a 3-box representation. In this paper we prove that every graph embedded in a fixed orientable surface, without short non-contractible cycles, has a 5-box representation. This directly implies that there is a function $f$, such that in every graph of genus $g$, a set of at most $f(g)$ vertices can be removed so that the resulting graph has a 5-box representation. We show that such a function $f$ can be made linear in $g$. Finally, we prove that for any proper minor-closed class $\mathcal{F}$, there is a constant $c(\mathcal{F})$ such that every graph of $\mathcal{F}$ without cycles of length less than $c(\mathcal{F})$ has a 3-box representation, which is best possible.Comment: 16 pages, 6 figures - revised versio

Three-coloring triangle-free graphs on surfaces II. 4-critical graphs in a disk

Let G be a plane graph of girth at least five. We show that if there exists a 3-coloring phi of a cycle C of G that does not extend to a 3-coloring of G, then G has a subgraph H on O(|C|) vertices that also has no 3-coloring extending phi. This is asymptotically best possible and improves a previous bound of Thomassen. In the next paper of the series we will use this result and the attendant theory to prove a generalization to graphs on surfaces with several precolored cycles.Comment: 48 pages, 4 figures This version: Revised according to reviewer comment