Kempe Equivalence of Colourings of Cubic Graphs

On a conjecture of Mohar concerning Kempe equivalence of regular graphs

Let G be a graph with a vertex colouring α. Let a and b be two colours. Then a connected component of the subgraph induced by those vertices coloured either a or b is known as a Kempe chain. A colouring of G obtained from α by swapping the colours on the vertices of a Kempe chain is said to have been obtained by a Kempe change. Two colourings of G are Kempe equivalent if one can be obtained fro...

Counting edge-Kempe-equivalence classes for 3-edge-colored cubic graphs

Two edge colorings of a graph are edge-Kempe equivalent if one can be obtained from the other by a series of edge-Kempe switches. This work gives some results for the number of edge-Kempe equivalence classes for cubic graphs. In particular we show every 2-connected planar bipartite cubic graph has exactly one edge-Kempe equivalence class. Additionally, we exhibit infinite families of nonplanar ...

ON THE EDGE COVER POLYNOMIAL OF CERTAIN GRAPHS

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...

Edge-colourings of cubic graphs admitting a solvable vertex-transitive group of automorphisms

Kempe Equivalence of Colorings

Several basic theorems about the chromatic number of graphs can be extended to results in which, in addition to the existence of a kcoloring, it is also shown that all k-colorings of the graph in question are Kempe equivalent. Here, it is also proved that for a planar graph with chromatic number less than k, all k-colorings are Kempe equivalent.