Polynomial bounds for chromatic number. III. Excluding a double star

Star chromatic number

A generalization of the chromatic number of a graph is introduced such that the colors are integers modulo n, and the colors on adjacent vertices are required to be as far apart as possible.

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 outer-independent double Italian domination number

An outer-independent double Italian dominating function (OIDIDF)on a graph $G$ with vertex set $V(G)$ is a function$f:V(G)longrightarrow {0,1,2,3}$ such that if $f(v)in{0,1}$ for a vertex $vin V(G)$ then $sum_{uin N[v]}f(u)geq3$,and the set $ {uin V(G)|f(u)=0}$ is independent. The weight ofan OIDIDF $f$ is the value $w(f)=sum_{vin V(G)}f(v)$. Theminimum weight of an OIDIDF on a graph $G$ is cal...

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