A graph Γ is symmetric if its automorphism group acts transitively on the arcs of Γ, and s-regular if its automorphism group acts regularly on the set of s-arcs of Γ. Tutte (1947, 1959) showed that every cubic finite symmetric cubic graph is s-regular for some s ≤ 5. Djokovič and Miller (1980) proved that there are seven types of arc-transitive group action on finite cubic graphs, characterised...

We characterise the quartic (i.e. 4-regular) multigraphs with the property that every edge lies in a triangle. The main result is that such graphs are either squares of cycles, line multigraphs of cubic multigraphs, or are obtained from the line multigraphs of cubic multigraphs by a number of simple subgraph-replacement operations. A corollary of this is that a simple quartic graph with every e...

If G and H are two cubic graphs, then we write H ≺ G, if G admits a proper edgecoloring f with edges of H , such that for each vertex x of G, there is a vertex y of H with f(∂G(x)) = ∂H(y). Let P and S be the Petersen graph and the Sylvester graph, respectively. In this paper, we introduce the Sylvester coloring conjecture. Moreover, we show that if G is a connected bridgeless cubic graph with ...

Using covering graph techniques, a structural result about connected cubic simple graphs admitting an edge-transitive solvable group of automorphisms is proved. This implies, among other, that every such graph can be obtained from either the 3-dipole Dip3 or the complete graph K4, by a sequence of elementary-abelian covers. Another consequence of the main structural result is that the action of...

If G and H are two cubic graphs, then we write H ≺ G if G admits a proper edge-coloring f with edges of H, such that for each vertex x of G, there is a vertex y of H with f(∂G(x)) = ∂H(y). Let P and S be the Petersen graph and the Sylvester graph, respectively. In this paper, we introduce the Sylvester coloring conjecture. Moreover, we show that if G is a connected bridgeless cubic graph with G...

The Fisher transformation acts on cubic graphs by replacing each vertex by a triangle. We explore the action of the Fisher transformation on the set of self-avoiding walks of a cubic graph. Iteration of the transformation yields a sequence of graphs with common critical exponents, and with connective constants converging geometrically to the golden mean. We consider the application of the Fishe...

Let $G$ be a simple graph of order $n$ and size $m$.The edge covering of $G$ is a set of edges such that every vertex of $G$ is incident to at least one edge of the set. The edge cover polynomial of $G$ is the polynomial$E(G,x)=sum_{i=rho(G)}^{m} e(G,i) x^{i}$,where $e(G,i)$ is the number of edge coverings of $G$ of size $i$, and$rho(G)$ is the edge covering number of $G$. In this paper we stud...

We show that if G is a connected bridgeless cubic graph whose every 2-factor is comprised of cycles of length five then G is the Petersen graph. ”The Petersen graph is an obstruction to many properties in graph theory, and often is, or conjectured to be, the only obstruction”. This phrase is taken from one of the series of papers by Robertson, Sanders, Seymour and Thomas that is devoted to the ...

A tricirculant is a graph admitting a non-identity automorphism having three cycles of equal length in its cycle decomposition. A graph is said to be symmetric if its automorphism group acts transitively on the set of its arcs. In this paper it is shown that the complete bipartite graph K3,3, the Pappus graph, Tutte’s 8-cage and the unique cubic symmetric graph of order 54 are the only connecte...

In this paper we develop the technique of a distribution decomposition for a graph. A formula is given to determine genus distribution of a cubic graph. Given any connected graph, it is proved that its genus distribution is the sum of those for some cubic graphs by using the technique.