On fans in multigraphs

We introduce a unifying framework for studying edge-colouring problems on multigraphs. This is defined in terms of a rooted directed multigraph F which is naturally associated to the set of fans based at a given vertex u in a multigraph G. We call F the “Fan Digraph”. We show that fans in G based at u are in one-to-one correspondence with directed trails in F starting at the root of F . We state and prove a central theorem about the fan digraph, which embodies many edge-colouring results and expresses them at a higher level of abstraction. Using this result, we derive short proofs of classical theorems. We conclude with a new, generalized version of Vizing’s Adjacency Lemma for multigraphs, which is stronger than all those known to the author. If S is a set, we let S and S denote, respectively, the set of ordered and unordered pairs of elements of S. An ordered pair will be enclosed by round brackets, e.g. (x, y) is an ordered pair, and an unordered pair will be enclosed by curly brackets, e.g. {x, y} is an unordered pair. If s ∈ S, and s = (x, y), the symbol s will be used to denote the ordered pair (y, x). The cardinality of S will be denoted by |S|. If θ is a function defined on a domain S, and T ⊂ S, the notation θ|T will be used to denote the restriction of θ to T , i.e. the function θ1 defined by θ1(t) = θ(t) for all t ∈ T. Similarly, if θ has domain S, codomain W and range R ⊂ W , and if R ⊂ W1 ⊂ W, it will be convenient to denote by θ| W1 ∗This paper has been published in the Journal of Graph Theory. Please cite this paper as: D. Cariolaro, On fans in multigraphs, Journal of Graph Theory, 51 (2006), 301-318. The present affiliation of the author is: Institute of Mathematics, Academia Sinica, Taiwan. E-mail address: [email protected]