6 research outputs found
Some results on the maximal chromatic polynomials of -connected -chromatic graphs
In 2015, Brown and Erey conjectured that every -connected graph on
vertices with chromatic number has at most
proper -colorings for all
. Engbers, Erey, Fox, and He proved this conjecture for . In this
paper, we prove Brown and Erey's conjecture under the condition that either the
clique number of is , or the independent number of is .Comment: 26 pages, 8 figures. Comments welcome
Tomescu\u27s Graph Coloring Conjecture for -Connected Graphs
Let PG(k) be the number of proper k-colorings of a finite simple graph G. Tomescu\u27s conjecture, which was recently solved by Fox, He, and Manners, states that PG(k)k!(k-1)(n β k) for all connected graphs G on n vertices with chromatic number kβ₯4. In this paper, we study the same problem with the additional constraint that G is β-connected. For 2-connected graphs G, we prove a tight bound PG(k)β€(k β 1)!((k β 1)(n β k+1) + ( - 1)n β k) and show that equality is only achieved if G is a k-clique with an ear attached. For ββ₯3, we prove an asymptotically tight upper bound PG(k)β€k!(k-1)n-l-k+1+O((k β 2)n ) and provide a matching lower bound construction. For the ranges kβ₯β or β β₯ (k-2)(k-1)+ 1 we further find the unique graph maximizing . We also consider generalizing β-connected graphs to connected graphs with minimum degree Ξ΄