Let be a connected Cayley graph of group G, then Γ is called normal if the right regular representation of G is a normal subgroup of , the full automorphism group of Γ. For the case where G is a finite nonabelian simple group and Γ is symmetric cubic Cayley graph, Caiheng Li and Shangjin Xu proved that Γ is normal with only two exceptions. Since then, the normality of nonsymmetric cubic Cayley ...

We estimate the minimum number of vertices of a cubic graph with given oddness and cyclic connectivity. We prove that a bridgeless cubic graph G with oddness ω(G) other than the Petersen graph has at least 5.41ω(G) vertices, and for each integer k with 2 6 k 6 6 we construct an infinite family of cubic graphs with cyclic connectivity k and small oddness ratio |V (G)|/ω(G). In particular, for cy...

Let S = {K1,3,K3, P4} be the set of connected graphs of size 3. We study the problem of partitioning the edge set of a graph G into graphs taken from any non-empty S′ ⊆ S. The problem is known to be NP-complete for any possible choice of S′ in general graphs. In this paper, we assume that the input graph is cubic, and study the computational complexity of the problem of partitioning its edge se...

The Berge-Fulkerson Conjecture states that every cubic bridgeless graph has six perfect matchings such that every edge of the graph is in exactly two of the perfect matchings. If the Berge-Fulkerson Conjecture is true, then what can we say about the proportion of edges of a cubic bridgeless graph that can be covered by k of its perfect matchings? This is the question we address in this paper. W...

A perfectly one-factorable (P1F) regular graph is a graph admitting a partition of the edge-set into one-factors such that the union of any two of them is a Hamiltonian cycle. The case of cubic graphs is treated. The existence of a P1F cubic graph is guaranteed for each admissible value of the number of vertices. A description of this class was obtained by Kotzig in 1962. It is the purpose of t...

A dominating set in a graph G is a set S of vertices such that every vertex outside S has a neighbor in S; the domination number γ(G) is the minimum size of such a set. The independent domination number, written i(G), is the minimum size of a dominating set that also induces no edges. Henning and Southey conjectured that if G is a connected cubic graph with sufficiently many vertices, then i(G)...

1 Petersen’s Theorem Recall that a graph is cubic if every vertex has degree exactly 3, and bridgeless if it cannot be disconnected by deleting any one edge (i.e., 2-edge-connected). Theorem 1 (Petersen) Any bridgeless cubic graph has a perfect matching. Proof: We will show that for any V ⊆ U , we have o(G−U) ≤ |U | (here o(G) is the number of odd components of the graph G). The theorem will th...

A cycle C in a graph is called stable if there exist no other cycle D in the same graph such that V (C) ⊆ V (D). In this paper we study stable cycles in snarks and based on our findings we are able to show that if a cubic graph G has a cycle of length at least |V (G)| − 9 then it has a cycle double cover. We also give a construction for an infinite snark family with stable cycles of constant le...

Lovász and Plummer conjectured, in the mid 1970’s, that every cubic graph G with no cutedge has an exponential in |V (G)| number of perfect matchings. In this work we show that every cubic planar graph G whose geometric dual graph is a stack triangulation has at least 3φ (G)|/72 distinct perfect matchings, where φ is the golden ratio. Our work builds on a novel approach relating Lovász and Plum...

The rainbowness, rb(G), of a connected plane graph G is the minimum number k such that any colouring of vertices of the graph G using at least k colours involves a face all vertices of which have different colours. For a cubic polyhedral (i.e. 3-connected plane) graph G we prove that n 2 + α∗ 1 − 1 ≤ rb(G) ≤ n− α∗ 0 + 1 , where α∗ 0 and α ∗ 1 denote the independence number and the edge independ...