6 research outputs found

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

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    In 2015, Brown and Erey conjectured that every 22-connected graph GG on nn vertices with chromatic number kβ‰₯4k\geq 4 has at most (xβˆ’1)kβˆ’1((xβˆ’1)nβˆ’k+1+(βˆ’1)nβˆ’k)(x-1)_{k-1}\big((x-1)^{n-k+1}+(-1)^{n-k}\big) proper xx-colorings for all xβ‰₯kx\geq k. Engbers, Erey, Fox, and He proved this conjecture for x=kx=k. In this paper, we prove Brown and Erey's conjecture under the condition that either the clique number of GG is kk, or the independent number of GG is 22.Comment: 26 pages, 8 figures. Comments welcome

    Tomescu\u27s Graph Coloring Conjecture for -Connected Graphs

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    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 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)≀(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 β„“β‰₯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 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΄
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