A Note on Forbidding Clique Immersions

Abstract

Robertson and Seymour proved that the relation of graph immersion is well-quasi-ordered for finite graphs. Their proof uses the results of graph minors theory. Surprisingly, there is a very short proof of the corresponding rough structure theorem for graphs without K timmersions; it is based on the Gomory-Hu theorem. The same proof also works to establish a rough structure theorem for Eulerian digraphs without K t -immersions, where K t denotes the bidirected complete digraph of order t. * mdevos@sfu.ca. Supported in part by an NSERC Discovery Grant (Canada) and a Sloan Fellowship. † jessica mcdonald@sfu.ca. Supported by an NSERC Postdoctoral Fellowship (Canada) ‡ mohar@sfu.ca.

Immersions in Highly Edge Connected Graphs

“…There is an easy structure theorem for graphs excluding a fixed H as an immersion [15], [4]. If we fix the graph H and let ∆ be the maximum degree of a vertex in H, then one obvious obstruction to a graph G containing H as an immersion is if every vertex of G has degree less than ∆.…”

Improper colourings inspired by Hadwiger's conjecture

“…Theorem 32 (following DeVos et al [10]). For every graph H with t vertices and every graph G that does not contain H as an immersion, there is a tree T and a T -partition of G with adhesion less than (t − 1) 2 , such that each bag has at most t − 1 vertices.…”

“…Since every graph with maximum degree at most t − 2 contains no (strong or weak) K t immersion, the clustered chromatic number of graphs excluding a (strong or weak) K t immersion is at least 1 4 (t + 4) . The proof of Theorem 30 uses the following structure theorem from DeVos et al [10]. The theorem is not explicitly proved in [10], but can be derived easily from the proof of [10, Theorem 1] on page 4 of that paper.…”

Nonrepetitive colorings of graphs excluding a fixed immersion or topological minor

“…A graph G contains a graph H as an immersion if the vertices of H can be mapped to distinct vertices of G, and the edges of H can be mapped to pairwise edge-disjoint paths in G, such that each edge vw of H is mapped to a path in G whose endpoints are the images of v and w. The image in G of each vertex in H is called a branch vertex. Structural and coloring properties of graphs excluding a fixed immersion have been widely studied [1,13,14,[18][19][20][22][23][24]34,36,40,42]. We prove that graphs excluding a fixed immersion have bounded nonrepetitive chromatic number.…”