A Vertex-Splitting Lemma, de Werra's Theorem, and Improper List Colourings

Abstract

We prove a new vertex-splitting lemma which states that if a multigraphGhas maximum multiplicity of at mostp, then each vertex u can be split into ⌈(d(u)/p)⌉ new vertices, ⌊(d(u)/p)⌋ of degreep, with the multiple edges being shared out between the new vertices in such a way that each multiple edge remains intact at at least one of its two endpoints. We apply this lemma to deduce very simply the theorem of de Werra that each bipartite multigraph has a balancedk-edge-colouring for each positive integerk. We also apply the lemma to improve two earlier theorems of the first author which state thatχ′^ℓ_s(G)=⌈(Δ(G)/s)⌉ if either (1)Gis a bipartite multigraph, or (2)sis even andGis any multigraph; hereχ′^ℓ_s(G) is thes-improper list chromatic index of G, i.e., the least possible value ofksuch that, if each edge ofGis provided with a list ℓ(e) of at leastkcolours, thenGhas ans-improper edge list colouring (so that at each vertexνnot more than s edges incident withνreceive the same colour), and, on each edge, the colour used lies in the list for that edge. The improvements that we obtain include that for each multiple edge of multiplicitym=m_G(u, ν), no colour is used on more than ⌈m_G(u, ν)/⌈(Δ(G)/s)⌉⌉+1 edges joininguandν.