Multigraphs with ∆ ≥ 3 are Totally - ( 2 ∆ - 1 ) - Choosable

The total graph T (G) of a multigraph G has as its vertices the set of edges and vertices of G and has an edge between two vertices if their corresponding elements are either adjacent or incident in G. We show that if G has maximum degree ∆(G), then T (G) is (2∆(G) − 1)-choosable. We give a linear-time algorithm that produces such a coloring. The best previous general upper bound for ∆(G) > 3 was ⌊ 3 2 ∆(G) + 2⌋, by Borodin et al. When ∆(G) = 4, our algorithm gives a better upper bound. When ∆(G) ∈ {3, 5, 6}, our algorithm matches the best known bound. However, because our algorithm is significantly simpler, it runs in linear time (unlike the algorithm of Borodin et al.). Throughout this paper, G is a connected multigraph (if G is not connected, we can color each component independently). For convenience, we refer to edges and vertices as elements of the graph. The total graph T (G) of a graph G has as its vertices the set of edges and vertices of G and has an edge between two vertices if their corresponding elements are either adjacent or incident in G. Let L be an assignment of lists to the vertices of a graph G. If G has a proper coloring such that each vertex v gets a color from its list L(v), then we say that G has an L-coloring. If G always has an L-coloring when each vertex has a list of size k, then we say that G is k-list-colorable (or k-choosable). In this paper, we study the problem of list-coloring a total graph. If a total graph T (G) is k-list-colorable, we say that G is totally-k-list-colorable (or totally-kchoosable). Often, our algorithm will greedily color all but a few edges and vertices of G; we generally call this uncolored subgraph H . This motivates the following definition. For a graph G and a subgraph H , we use G − H to mean G − (V (H) ∪ E(H)) (thus, edge uv may be present in G − H even if one or both of vertices u and v are missing). We say a graph algorithm runs in linear time if for fixed maximum degree the algorithm runs in time linear in the number of vertices of the graph. Let ∆(G) denote the maximum vertex degree of a graph G. Juvan et al. [4] showed that if ∆(G) = 3, then G is totally-5-choosable. Skulrattanakulchai and Gabow [5] used these ideas to show that if ∆(G) = 3, then we can construct a total-5-list-coloring in linear time. In this paper, we extend these ideas further to show that if ∆(G) ≥ 3, then we can construct a total-(2∆(G) − 1)-list-coloring in linear time. The best previous upper bound for ∆(G) > 3 was ⌊ 3 2 ∆(G)+2⌋, by Borodin et al. [1]. When ∆(G) = 4, our algorithm gives a new upper bound. When ∆(G) ∈ {3, 5, 6}, our algorithm matches the best known bound. However, because our algorithm is significantly simpler, it runs in linear time (unlike the algorithm of Borodin et al.). In Lemma 1, we show how to greedily construct a total-(2∆(G)− 1)-coloring for almost all of G. The rest of this paper shows that we can extend the coloring to all of G. University of Illinois, Urbana-Champaign