### The thickness and chromatic number of r-inflated graphs

Michael O. Albertson, Debra L. Boutin, Ellen Gethner
2010 Discrete Mathematics
Dedicated to Carsten Thomassen on the occasion of his 60th birthday. Abstract A graph has thickness t if the edges can be decomposed into t and no fewer planar layers. We study one aspect of a generalization of Ringel's famous Earth-Moon problem: what is the largest chromatic number of any thickness-2 graph? In particular, given a graph G we consider the r-inflation of G and find bounds on both the thickness and the chromatic number of the inflated graphs. In some instances the best possible
more » ... nds on both the chromatic number and thickness are achieved. We end with several open problems. Any edge of G (along with its incident vertices) induces a In particular, for any complete graph K s and any positive integer t, we have K s [t] = K st . Independence is invariant under inflation. That is, if the independence number of G is α then the independence number of G[r] is α as well. 5. If the clique number of G is ω, then the clique number of G[r] is rω. The Thickness of Cloned Graphs In this section we will begin the study of cloned graphs. In the next section we will generalize to r-inflated graphs. Example 1. Let P 3 be the path of length two with vertices labeled u, v, w in linear order. Denote the vertices of the clone of P 3 by u 1 , u 2 , v 1 , v 2 , w 1 , w 2 . Two plane drawings of the clone of P 3 are given in Figure 1 . The righthand drawing in Figure 1 , a straightline drawing given by nesting non-convex K 4 s, helps to illustrate the inductive argument that we will use next.