On the generalised colouring numbers of graphs that exclude a fixed minor

Abstract

The colouring number col(G) of a graph G is the minimum integer k such that there exists a linear ordering of the vertices of G in which each vertex v has back-degree at most k, i.e. v has at most k neighbours u with u<v. The colouring number is a structural measure that measures the edge density of subgraphs of G. For r≥1, the numbers colr(G) and wcolr(G) generalise the colouring number, where col1(G) and wcol1(G) are equivalent to col(G). For increasing values of r these measures converge to the well-known structural measures tree-width and tree-depth. For an n-vertex graph, coln(G) is equal to the tree-width of G and wcoln(G) is equal to the tree-depth of G.We show that if G excludes Kt as a minor, then colr(G)≤(t2)⋅(2r+1) and wcolr(G)≤(t2)r⋅(2r+1).It is easily observed that if G is planar, then colr(G)≤5r+3. The technically most demanding part of the paper is to show that for those graphs, wcolr(G)≤5r5. These results generalise to bounded genus graphs, i.e. if G is of genus g, then colr(G)≤(2g+3)(2r+1) and wcolr(G)≤2g(2r+1)+5r5.