A proper edge-coloring of a graph G is a mapping α : E(G)→ N such that α(e) 6= α(e′) for every pair of adjacent edges e, e′ ∈ E(G). A proper edge-coloring of a graph G is an interval coloring if the colors of edges incident to each vertex of G form an interval of integers. A graph G is interval colorable if it has an interval coloring. Since not all graphs admit an interval coloring, a measure ...

We consider the parameterised complexity of several list problems on graphs, with parameter treewidth or pathwidth. In particular, we show that List Edge Chromatic Number and List Total Chromatic Number are fixed parameter tractable, parameterised by treewidth, whereas List Hamilton Path is W[1]-hard, even parameterised by pathwidth. These results resolve two open questions of Fellows, Fomin, L...

B. Szegedy [Edge coloring models and reflection positivity, Journal of the American Mathematical Society 20 (2007) 969–988] showed that the number of homomorphisms into a weighted graph is equal to the partition function of a complex edge-coloring model. Using some results in geometric invariant theory, we characterize for which weighted graphs the edge-coloring model can be taken to be real va...

A proper edge coloring of a graph G is said to be acyclic if every cycle of G receives at least three colors. The acyclic edge chromatic number of G, denoted a(G), is the least number of colors in an acyclic edge coloring of G. Alon, Sudakov and Zaks [Acyclic edge coloring of graphs, J. Graph Theory 37 (2001), 157-167] conjectured that a(G) ≤ ∆(G) + 2 holds for any graph G. In present paper, we...

A multicircuit is a multigraph whose underlying simple graph is a circuit (a connected 2-regular graph). The List-Colouring Conjecture (LCC) is that every multigraph G has edgechoosability (list chromatic index) ch’(G) equal to its chromatic index x’(G). In this paper the LCC is proved first for multicircuits, and then, building on results of Peterson and Woodall, for any multigraph G in which ...

In this paper we consider two problems: the edge coloring and the strong edge coloring problems on unit disk graphs (UDGs). Both problems have important applications in wireless sensor networks as they can be used to model link scheduling problems in such networks. It is well known that both problems are NP-complete, and approximation algorithms for them have been extensively studied under the ...

An edge-coloring of a graph G1⁄4 ðV ; EÞ is a function c that assigns an integer c(e) (called color) in f 0;1;2;...g to every edge eAE so that adjacent edges are assigned different colors. An edge-coloring is compact if the colors of the edges incident to every vertex form a set of consecutive integers. The deficiency problem is to determine the minimum number of pendant edges that must be adde...