Extremal problems on variations of graph colorings
Ilkyoo Choi · 2014
This thesis investigates various coloring problems in graph theory. Graph coloring is an essential part of combinatorics and discrete mathematics, as it deals with the fundamental problem of partitioning objects so that each part satisfies a certain condition. In particular, we study how forbidding certain structures (subgraphs) affects a given coloring parameter. Open since 1977, the Borodin–Kostochka Conjecture states that given a graph G with maximum degree ∆(G) at least 9, if G has no clique of size ∆(G), then G is (∆(G) − 1)-colorable. The current best result by Reed shows that the statement of the Borodin– Kostochka Conjecture is true for graphs with maximum degree at least 1014. We produce a result of this type for the list chromatic number; namely, we prove that given a graph G with maximum degree at least 1020, if G has no clique of size ∆(G), then G is (∆(G) − 1)-choosable. Cai, Wang, and Zhu proved that a toroidal graph with no 6-cycles is 5-choosable, and they conjectured that the only case when it is not 4-choosable is when the graph contains K5. We disprove this conjecture by constructing a family of graphs containing neither 6-cycles nor K5 that are not even 4-colorable. This family is embeddable not only on a torus, but also on any surface except the plane and the projective plane. We prove a slightly weaker statement suggested by Zhu that toroidal graphs containing neither 6-cycles nor K−5 are 4-choosable. We also study questions regarding variants of coloring. We provide additional posi-tive support for a question by Škrekovski regarding choosability with separation for planar graphs, and we completely answer a question by Raspaud and Wang regarding vertex ar-ii boricity for toroidal graphs. We also improve results regarding improper coloring of planar graphs, responding to a question of Montassier and Ochem. iii Dedicated to my family, iv