The List Edge Colouring Conjecture asserts that, given any multigraph G with chromatic index k and any set system {Se : e C E(G)} with each [Se[ = k, we can choose elements Se E Se such that Ser whenever e and f are adjacent edges. Using a technique of Alon and Tarsi which involves the graph monomial ]-I{xu -Xv : uv E E} of an oriented graph, we verify this conjecture for certain families of 1-factorable multigraphs, including 1-factorable planar 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 ∆, it runs in linear time if ∆ is bounded by a constant, and in polynomial time if ∆ = O(3 log n). We mention various applications to areas including hypergraph matchings and radius k independent sets. We extend our results to the problem of determining whether a graph has a weighting which makes it well-covered.

This paper provides new upper bounds on the spectral radius \ (largest eigenvalue of the adjacency matrix) of graphs embeddable on a given compact surface. Our method is to bound the maximum rowsum in a polynomial of the adjacency matrix, using simple consequences of Euler's formula. Let # denote the Euler genus (the number of crosscaps plus twice the number of handles) of a fixed surface 7. Then (i) for n 3, every n-vertex graph embeddable on 7 has \ 2+-2n+8#&6, and (ii) a 4-connected graph with a spherical or 4-representative embedding on 7 has \ 1+-2n+2#&3. Result (i) is not sharp, as Guiduli and Hayes have recently proved that the maximum value of \ is 3Â2+-2n+o(1) as n Ä for graphs embeddable on a fixed surface. However, (i) is the only known bound that is computable, valid for all n 3, and asymptotic to -2n like the actual maximum value of \. Result (ii) is sharp for the sphere or plane (#=0), with equality holding if and only if the graph is a``double wheel'' 2K 1 +C n&2 (+ denotes join). For other surfaces we show that (ii) is within O(1Ân 1Â2 ) of sharpness. We also show that a recent bound on \ by Hong can be deduced by our method. Academic Press

In this paper we study connected (g,f )-factors. We describe an algorithm to connect together an arbitrary spanning subgraph of a graph, without increasing the vertex degrees too much; if the algorithm fails we obtain information regarding the structure of the graph. As a consequence we give sufficient conditions for a graph to have a connected (g,f )-factor, in terms of the number of components obtained when we delete a set of vertices. As corollaries we can derive results of Win [S. Win suffices. We also show that highly edge-connected graphs have spanning trees of relatively low degree; in particular, an m-edge-connected graph G has a spanning tree T such that deg T (v) 2 þ ddeg G (v)/me for each vertex v.

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