Hedetniemi’s Conjecture and Adjoint Functors in Thin Categories

We survey results on Hedetniemi’s conjecture which are connected to adjoint functors in the “thin” category of graphs, and expose the obstacles to extending these results

Tabular graphs and chromatic sum

AbstractThe chromatic sum of a graph is the minimum total of the colors on the vertices taken over all possible proper colorings using positive integers. Erdös et al [Graphs that require many colors to achieve their chromatic sum, Congr. Numer. 71 (1990) 17–28.] considered the question of finding graphs with minimum number of vertices that require t colors beyond their chromatic number k to obtain their chromatic sum. The number of vertices of such graphs is denoted by P(k,t). They presented some upper bounds for this parameter by introducing certain constructions. In this paper we give some lower bounds for P(k,t) and considerably improve the upper bounds by introducing a class of graphs, called tabular graphs. Finally, for fixed t and sufficiently large k the exact value of P(k,t) is determined

Some Concepts in List Coloring

In this paper uniquely list colorable graphs are studied. A graph G is said to be uniquely k--list colorable if it admits a k--list assignment from which G has a unique list coloring. The minimum k for which G is not uniquely k--list colorable is called the m--number of G. We show that every triangle--free uniquely colorable graph with chromatic number k + 1 is uniquely k--list colorable. A bound for the m--number of graphs is given, and using this bound it is shown that every planar graph has m--number at most 4. Also we introduce list criticality in graphs and characterize all 3--list critical graphs. It is conjectured that every # # # --critical graph is # # -- critical and the equivalence of this conjecture to the well known list coloring conjecture is shown

A Counterexample for Hilton-Johnson's Conjecture on List-Coloring of Graphs

In this paper a conjecture of A. Hilton and P. Johnson on list coloring of graphs is disproved. By modifying our counterexample, we also answer some other questions concerning Hall numbers.