Edge reductions in cyclically k-connected cubic graphs

AbstractThis paper examines edge reductions in cyclically k-connected cubic graphs, focusing on when they preserve the cyclic k-connectedness. For a cyclically k-connected cubic graph G, we denote by Nk(G) the set of edges whose reduction gives a cubic graph which is not cyclically k-connected. With the exception of three graphs, Nk(G) consists of the edges in independent k-edge cuts. For this reason we examine the properties and interactions between independent k-edge cuts in cyclically k-connected cubic graphs. These results lead to an understanding of the structure of G[Nk]. For every k, we prove that G[Nk] is a forest with at least k trees if G is a cyclically k-connected cubic graph with girth at least k + 1 and Nk ≠ ⊘. Let fk(ν) be the smallest integer such that |Nk(G)| ≤ fk(ν) for all cyclically k-connected cubic graphs G on ν vertices. For all cyclically 3-connected cubic graphs G such that 6 ≤ ν(G) and N3 ≠ ⊘, we prove that G[N3] is a forest with at least three trees. We determine f3 and state a characterization of the extremal graphs. We define a very restricted subset N4b of N4 and prove that if N4g = N4 − N4b ≠ ⊘, then G[N4g] is a forest with at least four trees. We determine f4 and state a characterization of the extremal graphs. There exist cyclically 5-connected cubic graphs such that E(G) = N5(G), for every ν such that 10 ≤ ν and 16 ≠ ν. We characterize these graphs. Let gk(ν) be the smallest integer such that |Nk(G)| ≤ gk(ν) for all cyclically k-connected cubic graphs G with ν vertices and girth at least k + 1. For k ∈ {3, 4, 5}, we determine gk and state a characterization of the extremal graphs

Nash Williams Conjecture and the Dominating Cycle Conjecture

The disproved Nash Williams conjecture states that every 4-regular 4-connected graph has a hamiltonian cycle. We show that a modification of this conjecture is equivalent to the Dominating Cycle Conjecture

Cyclically five-connected cubic graphs

A cubic graph $G$ is cyclically 5-connected if $G$ is simple, 3-connected, has at least 10 vertices and for every set $F$ of edges of size at most four, at most one component of $G\backslash F$ contains circuits. We prove that if $G$ and $H$ are cyclically 5-connected cubic graphs and $H$ topologically contains $G$, then either $G$ and $H$ are isomorphic, or (modulo well-described exceptions) there exists a cyclically 5-connected cubic graph $G'$ such that $H$ topologically contains $G'$ and $G'$ is obtained from $G$ in one of the following two ways. Either $G'$ is obtained from $G$ by subdividing two distinct edges of $G$ and joining the two new vertices by an edge, or $G'$ is obtained from $G$ by subdividing each edge of a circuit of length five and joining the new vertices by a matching to a new circuit of length five disjoint from $G$ in such a way that the cyclic orders of the two circuits agree. We prove a companion result, where by slightly increasing the connectivity of $H$ we are able to eliminate the second construction. We also prove versions of both of these results when $G$ is almost cyclically 5-connected in the sense that it satisfies the definition except for 4-edge cuts such that one side is a circuit of length four. In this case $G'$ is required to be almost cyclically 5-connected and to have fewer circuits of length four than $G$. In particular, if $G$ has at most one circuit of length four, then $G'$ is required to be cyclically 5-connected. However, in this more general setting the operations describing the possible graphs $G'$ are more complicated.Comment: 47 pages, 5 figures. Revised according to referee's comments. To appear in J. Combin. Theory Ser.

Excluded minors in cubic graphs

Let G be a cubic graph, with girth at least five, such that for every partition X,Y of its vertex set with |X|,|Y|>6 there are at least six edges between X and Y. We prove that if there is no homeomorphic embedding of the Petersen graph in G, and G is not one particular 20-vertex graph, then either G\v is planar for some vertex v, or G can be drawn with crossings in the plane, but with only two crossings, both on the infinite region. We also prove several other theorems of the same kind.Comment: 62 pages, 17 figure