2 research outputs found
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΄