Polynomial bounds for chromatic number II: Excluding a star‐forest

The locating-chromatic number for Halin graphs

Let G be a connected graph. Let f be a proper k -coloring of G and Π = (R_1, R_2, . . . , R_k) bean ordered partition of V (G) into color classes. For any vertex v of G, define the color code c_Π(v) of v with respect to Π to be a k -tuple (d(v, R_1), d(v, R_2), . . . , d(v, R_k)), where d(v, R_i) is the min{d(v, x)|x ∈ R_i}. If distinct vertices have distinct color codes, then we call f a locat...

Bounds on the Chromatic Polynomial and on the Number of Acyclic Orientations of a Graph

An upper bound is given on the number of acyclic orientations of a graph, in terms of the spectrum of its Laplacian. It is shown that this improves upon the previously known bound, which depended on the degree sequence of the graph. Estimates on the new bound are provided. A lower bound on the number of acyclic orientations of a graph is given, with the help of the probabilistic method. This ar...

Topological Lower Bounds for the Chromatic Number: A hierarchy

Topological lower bounds for the chromatic number: A hierarchy

Kneser’s conjecture, first proved by Lovász in 1978, states that the graph with all kelement subsets of {1, 2, . . . , n} as vertices and all pairs of disjoint sets as edges has chromatic number n−2k+2. Several other proofs have been published (by Bárány, Schrijver, Dol’nikov, Sarkaria, Kř́ıž, Greene, and others), all of them based on the Borsuk–Ulam theorem from algebraic topology, but otherwis...

Upper bounds for the chromatic number of a graph

For a connected graph G of order n, the clique number ω(G), the chromatic number χ(G) and the independence number α(G) satisfy ω(G) ≤ χ(G) ≤ n − α(G) + 1. We will show that the arithmetic mean of the previous lower and upper bound provides a new upper bound for the chromatic number of a graph.