Clique minors in double‐critical graphs

• Published in: Journal of graph theory Vol. 88; no. 2; pp. 347 - 355
• Main Authors: Rolek, Martin, Song, Zi‐Xia
• Format: Journal Article
• Language: English
• Published: Hoboken Wiley Subscription Services, Inc 01.06.2018
• Subjects: clique minor; double‐critical graph; Graph coloring; Graphs; separating set
• ISSN: 0364-9024; 1097-0118
• DOI: 10.1002/jgt.22216

A connected t‐chromatic graph G is double‐critical if G−{u,v} is (t−2)‐colorable for each edge uv∈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'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 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.