A special case of Hadwiger's conjecture

Abstract

We investigate Hadwiger's conjecture for graphs with no stable set of size 3. Such a graph on at least 2t − 1 vertices is not t − 1 colorable, so is conjectured to have a K t minor. There is a strengthening of Hadwiger's conjecture in this case, which states that there is a K t minor in which the preimage of each vertex of K t is a single vertex or an edge. We prove this strengthened version for graphs with an even number of vertices and fractional clique covering number less than 3. We investigate several possible generalizations and obtain counterexamples for some and improved results from others. We also show that for sufficiently large n, a graph on n vertices with no stable set of size 3 has a K 1 9 n 4/5 minor using only vertices and single edges as preimages of vertices.

An approximate version of Hadwiger's conjecture for claw‐free graphs

“…One thing to note is that all graphs with = 2 are claw-free and there has been much research and multiple papers written on the subject of Hadwiger's conjecture for this class of graphs with only minimal progress [4,8]. So the next best thing would be to prove Hadwiger's conjecture for all claw-free graphs with >2.…”

“…Then some anticomponent, say C, of G contains a triad {v 1 , v 2 , v 3 } (since anticomponents are complete to each other by definition). 4 } is a claw, a contradiction. This proves the claim.…”

Complete graph immersions in dense graphs

“…Finally, we focus on the study of a special class of graphs, the graphs G that have no independent set of size three, or equivalently, whose independence number α(G) is at most 2. This class of graphs has been extensively studied in an attempt to solve Hadwiger's conjecture (see [2,5,6,20]). It is for this reason that we are interested in Abu-Khzam and Langston's conjecture restricted to these graphs.…”

Connected matchings in special families of graphs.

“…Otherwise, [3], the author shows that any n = 2 graph with connectivity less than n/2 satisfies SSH. We show the following for higher connectivity.…”

“…Taken altogether, the results of this section show that a counterexample to Conjecture 5 must be highly connected, yet avoid large cliques. As Blasiak remarks in [3], this means that the most "mysterious" cases arise when minimum degree and connectivity are close to the number of vertices in the graph. These are what we call the small clique cases.…”