Mark N. Ellingham

List edge colourings of some 1-factorable multigraphs

AbstractThe List Edge Colouring Conjecture asserts that, given any multigraphG with chromatic indexk and any set system {Se:e∈E(G)} with each |Se|=k, we can choose elementsse∈Sesuch thatse≠sfwhenevere andf are adjacent edges. Using a technique of Alon and Tarsi which involves the graph monomial

Local Structure When All Maximal Independent Sets Have Equal Weight

\prod {\left\{ {xu - x_\upsilon :u\upsilon \in E} \right\}}

Spanning Planar Subgraphs of Graphs in the Torus and Klein Bottle

of an oriented graph, we verify this conjecture for certain families of 1-factorable multigraphs, including 1-factorable planar graphs.

P 3 -isomorphisms for graphs

A graph is t-tough if the number of components of G\S is at most |S|-t for every cutset S ⊆ V (G). A k-walk in a graph is a spanning closed walk using each vertex at most k times. When k = 1, a 1-walk is a Hamilton cycle, and a longstanding conjecture by Chvatal is that every sufficiently tough graph has a 1-walk. When k ≥ 3, Jackson and Wormald used a result of Win to show that every sufficiently tough graph has a k-walk. We fill in the gap between k = 1 and k ≥ 3 by showing that, when k = 2, every sufficiently tough (specifically, 4-tough) graph has a 2-walk. To do this we first provide a new proof for and generalize a result by Win on the existence of a k-tree, a spanning tree with every vertex of degree at most k. We also provide new examples of tough graphs with no k-walk for k ≥ 2.

Non-hamiltonian 3-connected cubic bipartite graphs

In many combinatorial situations there is a notion of independence of a set of points. Maximal independent sets can be easily constructed by a greedy algorithm, and it is of interest to determine, for example, if they all have the same size or the same parity. Both of these questions may be formulated by weighting the points with elements of an abelian group, and asking whether all maximal independent sets have equal weight. If a set is independent precisely when its elements are pairwise independent, a graph can be used as a model. The question then becomes whether a graph, with its vertices weighted by elements of an abelian group, is well-covered, i.e., has all maximal independent sets of vertices with equal weight. This problem is known to be co-NP-complete in general. We show that whether a graph is well-covered or not depends on its local structure. Based on this, we develop an algorithm to recognize well-covered graphs. For graphs with n vertices and maximum degree

Cycles through ten vertices in 3-connected cubic graphs

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The nonorientable genus of complete tripartite graphs

, it runs in linear time if

Bidegreed graphs are edge reconstructible

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