A Tight Linear Bound to the Chromatic Number of $$(P_5, K_1+(K_1\cup K_3))$$-Free Graphs

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...

The locating chromatic number of the join of graphs

‎Let $f$ be a proper $k$-coloring of a connected graph $G$ and‎ ‎$Pi=(V_1,V_2,ldots,V_k)$ be an ordered partition of $V(G)$ into‎ ‎the resulting color classes‎. ‎For a vertex $v$ of $G$‎, ‎the color‎ ‎code of $v$ with respect to $Pi$ is defined to be the ordered‎ ‎$k$-tuple $c_{{}_Pi}(v)=(d(v,V_1),d(v,V_2),ldots,d(v,V_k))$‎, ‎where $d(v,V_i)=min{d(v,x):~xin V_i}‎, ‎1leq ileq k$‎. ‎If‎ ‎distinct...

A tight bound on the set chromatic number

The purpose of this note is to provide a tight bound on the set chromatic number of a graph in terms of its chromatic number. Namely, for all graphs G, we show that χs(G) > dlog2 χ(G)e+ 1, where χs(G) and χ(G) are the set chromatic number and the chromatic number of G, respectively. This answers in the affirmative a conjecture of Gera, Okamoto, Rasmussen and Zhang.

A lower bound on the chromatic number of Mycielski graphs

In this work we give a new lower bound on the chromatic number of a Mycielski graph Mi. The result exploits a mapping between the coloring problem and a multiprocessor task scheduling problem. The tightness of the bound is proved for i = 1; : : : ; 8. c © 2001 Elsevier Science B.V. All rights reserved.

A bound for the game chromatic number of graphs