Douglas R. Woodall

List Edge and List Total Colourings of Multigraphs

This paper exploits the remarkable new method of Galvin (J. Combin. Theory Ser. B63(1995), 153?158), who proved that the list edge chromatic number??list(G) of a bipartite multigraphGequals its edge chromatic number??(G). It is now proved here that if every edgee=uwof a bipartite multigraphGis assigned a list of at least max{d(u),d(w)} colours, thenGcan be edge-coloured with each edge receiving a colour from its list. If every edgee=uwin an arbitrary multigraphGis assigned a list of at least max{d(u),d(w)}+?12min{d(u),d(w)}? colours, then the same holds; in particular, ifGhas maximum degree?=?(G) then??list(G)??32??. Sufficient conditions are given in terms of the maximum degree and maximum average degree ofGin order that??list(G)=?and??list(G)=?+1. Consequences are deduced for planar graphs in terms of their maximum degree and girth, and it is also proved that ifGis a simple planar graph and??12 then??list(G)=?and??list(G)=?+1.

Defective colorings of graphs in surfaces: Partitions into subgraphs of bounded valency

We call a graph (m, k)-colorable if its vertices can be colored with m colors in such a way that each vertex is adjacent to at most k vertices of the same color as itself. For the class of planar graphs, and the class of outerplanar graphs, we determine all pairs (m, k) such that every graph in the class is (m, k)-colorable. We include an elementary proof (not assuming the truth of the four-color theorem) that every planar graph is (4, 1)-colorable. Finally, we prove that, for each compact surface S, there is an integer k = k(S) such that every graph in S can be (4, k)-colored; we conjecture that 4 can be replaced by 3 in this statement.

The binding number of a graph and its Anderson number

Abstract The binding number of a graph G, bind(G), is defined; some examples of its calculation are given, and some upper bounds for it are proved. It is then proved that, if bind(G) ≥ c, then G contains at least |G| c (c + 1) disjoint edges if 0 ≤ c ≤ 1 2 , at least | G | (3c − 2) 3c − 2(c − 1) c disjoint edges if 1 ≤ c ≤ 4 3 , a Hamiltonian circuit if c ≥ 3 2 , and a circuit of length at least 3(| G | −1)(c − 1) c if 1 3 2 and G is not one of two specified exceptional graphs. Each of these results is best possible. The Anderson number of a graph is defined. The Anderson numbers of a few very simple graphs are determined; and some rather weak bounds are obtained, and some conjectures made, on the Anderson numbers of graphs in general.

Distances Realized by Sets Covering the Plane

Abstract A proof is given of the (known) result that, if real n-dimensional Euclidean space Rn is covered by any n + 1 sets, then at least one of these sets is such that each distance d(0

Total colorings of planar graphs with large maximum degree

It is proved that a planar graph with maximum degree Δ ≥ 11 has total (vertex-edge) chromatic number

ACYCLIC COLOURINGS OF PLANAR GRAPHS WITH LARGE GIRTH

Delta; + 1.

Cyclic-order graphs and Zarankiewicz's crossing-number conjecture

A proper vertex-colouring of a graph is acyclic if there are no 2-coloured cycles. It is known that every planar graph is acyclically 5-colourable, and that there are planar graphs with acyclic chromatic number v a fl 5 and girth g fl 4. It is proved here that a planar graph satisfies v a % 4i fg & 5 and v a % 3i fg & 7.

Dividing a cake fairly

Zarankiewiczs conjecture, that the crossing number of the completebipartite graph K,,,, is [

Choosability conjectures and multicircuits

rnllfr (m - 1)Jl; nj[

Electronic stability of fullerenes: eigenvalue theorems for leapfrog carbon clusters

(n - 1)j, was proved by Kleitman when min(rn, n) s 6, but was unsettled in all other cases. The cyclic-order graph CO, arises naturally in the study of this conjecture; it is a vertex-transitive harmonic diametrical (even) graph. In this paper the properties of cyclic-order graphs are investigated and used as the basis for computer programs that have verified Zarankiewiczs conjecture for K7,7 and K7,9; thus the smallest unsettled cases are now K7,11 and K9,9. 0 1993 John Wiley & Sons, Inc.