# Colouring series-parallel graphs

#### Abstract

In this paper we prove two results, which are not related to one another except that they both involve colouring series-parallel graphs. The easier of the two concerns the correctness of a "greedy algorithm" for finding a vertex-colouring of a seriesparallel graph, and we postpone all further discussion of this to Section 6. The other result concerns edge-colourings, and will be dealt with in Sections 1-5. A k-edge-colouring of a graph G is a map ~ : E(G) ~ {1, . . . , k} such that for distinct edges e, f if ~(e) = a ( f ) then e and f have no common end. (Graphs in this paper are finite, and may have multiple edges but not loops; V(G) and E(G) denote the vertexand edge-sets of a graph G.) The chromatic index xI(G) is the minimum k >_ 0 such that G has a k-edge-colouring. The problem of determining XII!G) is NP-hard; indeed, deciding whether a 3-connected cubic graph G satisfies X (G) = 3 or not is NP-complete [6]. On the other hand, for planar graphs G it is not known whether determining xI(G) is NP-hard; and indeed for planar graphs there is a conjectured minimax formula (which, if true, would be convertible into a polynomial algorithm, via [3,4]) which we now discuss. Let A(G) denote the maximum valency of the vertices of G (the valency of a vertex is the number of edges incident with it). Then clearly xt(G) >_ A(G), but equality need not occur (for example, when G = Ks). There is, however, another useful lower bound on Xt(G), as follows. For X C_ V(G), we denote by ~" the set of edges of G with both ends in X. In any xl(G)-edge-colouring ~ of G, at most []XI/2 j edges in X have the same colour (where ~pJ denotes the greatest integer n with n _< p), and hence IXI _< xI(G). [IXI/2J. This follows from our other bound

DOI: 10.1007/BF02128672

### Cite this paper

@article{Seymour1990ColouringSG, title={Colouring series-parallel graphs}, author={Paul D. Seymour}, journal={Combinatorica}, year={1990}, volume={10}, pages={379-392} }