Clique Minors in Double-critical Graphs

A connected $t$-chromatic graph $G$ is \dfn{double-critical} if $G \backslash\{u, v\}$ is $(t-2)$-colorable for each edge $uv\in E(G)$. A long standing conjecture of Erd\H{o}s and Lov\'asz that the complete graphs are the only double-critical $t$-chromatic graphs remains open for all $t\ge6$. Given the difficulty in settling Erd\H{o}s and Lov\'asz's conjecture and motivated by the well-known Hadwiger's conjecture, Kawarabayashi, Pedersen and Toft proposed a weaker conjecture that every double-critical $t$-chromatic graph contains a $K_t$ minor and verified their conjecture for $t\le7$. Albar and Gon\c{c}alves recently proved that every double-critical $8$-chromatic graph contains a $K_8$ minor, and their proof is computer-assisted. In this paper we prove that every double-critical $t$-chromatic graph contains a $K_t$ minor for all $t\le9$. Our proof for $t\le8$ is shorter and computer-free.Comment: 11 pages, to appear in J. Graph Theor

Coloring Graphs with Forbidden Minors

A graph H is a minor of a graph G if H can be obtained from a subgraph of G by contracting edges. My research is motivated by the famous Hadwiger\u27s Conjecture from 1943 which states that every graph with no Kt-minor is (t − 1)-colorable. This conjecture has been proved true for t ≤ 6, but remains open for all t ≥ 7. For t = 7, it is not even yet known if a graph with no K7-minor is 7-colorable. We begin by showing that every graph with no Kt-minor is (2t − 6)- colorable for t = 7, 8, 9, in the process giving a shorter and computer-free proof of the known results for t = 7, 8. We also show that this result extends to larger values of t if Mader\u27s bound for the extremal function for Kt-minors is true. Additionally, we show that any graph with no K−8 - minor is 9-colorable, and any graph with no K=8-minor is 8-colorable. The Kempe-chain method developed for our proofs of the above results may be of independent interest. We also use Mader\u27s H-Wege theorem to establish some sufficient conditions for a graph to contain a K8-minor. Another motivation for my research is a well-known conjecture of Erdos and Lovasz from 1968, the Double-Critical Graph Conjecture. A connected graph G is double-critical if for all edges xy ∈ E(G), χ(G−x−y) = χ(G)−2. Erdos and Lovasz conjectured that the only double-critical t-chromatic graph is the complete graph Kt. This conjecture has been show to be true for t ≤ 5 and remains open for t ≥ 6. It has further been shown that any non-complete, double-critical, t-chromatic graph contains Kt as a minor for t ≤ 8. We give a shorter proof of this result for t = 7, a computer-free proof for t = 8, and extend the result to show that G contains a K9-minor for all t ≥ 9. Finally, we show that the Double-Critical Graph Conjecture is true for double-critical graphs with chromatic number t ≤ 8 if such graphs are claw-free

The Extremal Function for K10 Minors

We prove that every graph on n >= 8 vertices and at least 8n-35 edges either has a K10 minor or is isomorphic to some graph included in a few families of exceptional graphs.Ph.D

Clique Minors In Double-Critical Graphs

A connected t-chromatic graph G is double-critical if G - { u,v } is (t-2) -colorable for each edge ∈ E(G). A long-standing conjecture of Erdős and Lovász that the complete graphs are the only double-critical t-chromatic graphs remains open for all t ≥ 6. Given the difficulty in settling Erdős and Lovász\u27s conjecture and motivated by the well-known Hadwiger\u27s conjecture, Kawarabayashi, Pedersen, and Toft proposed a weaker conjecture that every double-critical t-chromatic graph contains a Kt minor and verified their conjecture for t ≤ 7. Albar and Gonçalves recently proved that every double-critical 8-chromatic graph contains a K8 minor, and their proof is computer assisted. In this article, we prove that every double-critical t-chromatic graph contains a Kt minor for all t ≤ 9 Our proof for t ≤ 8 is shorter and computer free