Gaps in the Chromatic Spectrum of Face-Constrained Plane Graphs

Gaps in the Chromatic Spectrum of Face-Constrained Plane Graphs

Let G be a plane graph whose vertices are to be colored subject to constraints on some of the faces. There are 3 types of constraints: a C indicates that the face must contain two vertices of a Common color, a D that it must contain two vertices of a Different color and a B that Both conditions must hold simultaneously. A coloring of the vertices of G satisfying the facial constraints is a stri...

Maximum face-constrained coloring of plane graphs

Edge-face chromatic number and edge chromatic number of simple plane graphs

Given a simple plane graph G, an edge-face k-coloring of G is a function : E(G) [ F (G) ! f1, . . . ,kg such that, for any two adjacent or incident elements a, b 2 E(G) [ F (G), (a) 61⁄4 (b). Let e(G), ef(G), and (G) denote the edge chromatic number, the edge-face chromatic number, and the maximum degree of G, respectively. In this paper, we prove that ef(G) 1⁄4 e(G) 1⁄4 (G) for any 2-connected...

The chromatic number of ordered graphs with constrained conflict graphs

An ordered graph G is a graph whose vertices are the first n positive integers. The edges are interpreted as tuples (u, v) with u < v. For a positive integer s, a matrix M ∈ Zs×4, and a vector p = (p, . . . , p) ∈ Z we build a conflict graph whose vertices are the edges of G by saying that edges (u, v) and (x, y) are conflicting ifM(u, v, x, y)> > p orM(x, y, u, v)> > p, where the comparison is...

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...