Some results on the maximal chromatic polynomials of 22-connected kk-chromatic graphs

In 2015, Brown and Erey conjectured that every $2$-connected graph $G$ on $n$ vertices with chromatic number $k\geq 4$ has at most $(x-1)_{k-1}\big((x-1)^{n-k+1}+(-1)^{n-k}\big)$ proper $x$-colorings for all $x\geq k$. Engbers, Erey, Fox, and He proved this conjecture for $x=k$. In this paper, we prove Brown and Erey's conjecture under the condition that either the clique number of $G$ is $k$, or the independent number of $G$ is $2$.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 – k) for all connected graphs G on n vertices with chromatic number k≥4. In this paper, we study the same problem with the additional constraint that G is ℓ-connected. For 2-connected graphs G, we prove a tight bound PG(k)≤(k – 1)!((k – 1)(n – k+1) + ( - 1)n – k) and show that equality is only achieved if G is a k-clique with an ear attached. For ℓ≥3, we prove an asymptotically tight upper bound PG(k)≤k!(k-1)n-l-k+1+O((k – 2)n ) and provide a matching lower bound construction. For the ranges k≥ℓ or ℓ ≥ (k-2)(k-1)+ 1 we further find the unique graph maximizing . We also consider generalizing ℓ-connected graphs to connected graphs with minimum degree δ