We prove that if is a subgraph of complete multipartite graph , then contains connected component satisfying . use this to every 3-coloring the edges monochromatic with at least edges. further show such coloring has circuit fraction This verifies conjecture Conlon and Tyomkyn. Moreover, for general we -coloring

In this paper, we study the coloring problem for the undirected binary de Bruijn interconnection network. The coloring scheme is simple and fast. We propose the coloring algorithm by using the pseudo shortestpath spanning tree rooted at (0 00). Each processor can find its color number by its own identity. Then, based on our coloring algorithm, we propose a 1-fair alternator. Our design is optim...

The circular arc coloring problem is to find a minimum coloring of a set of arcs of a circle so that no two overlapping arcs share a color. This N P-hard problem arises in a rich variety of applications and has been studied extensively. In this paper we present an O(n2m) combinatorial algorithm for optimally coloring any set of arcs that corresponds to a perfect graph, and propose a new approac...

An acyclic edge coloring of a graph G is a proper edge coloring such that every cycle is colored with at least three colors. The acyclic chromatic index χa(G) of a graph G is the least number of colors in an acyclic edge coloring of G. It was conjectured that χa(G) ≤ ∆(G) + 2 for any simple graph G with maximum degree ∆(G). In this paper, we prove that every planar graph G admits an acyclic edg...

An equitable graph coloring is a proper vertex coloring, such that the size of the color classes differ by at most one. We present a flow-based scheme for generating pruning rules for enumerative algorithms such as DSATUR. The scheme models the extendability of a partial (equitable) coloring into an equitable coloring via network flows. Computational experiments show that significant reductions...

9 An acyclic coloring is a proper coloring with the additional property that the union of 10 any two color classes induces a forest. We show that every graph with maximum degree at 11 most 5 has an acyclic 7-coloring. We also show that every graph with maximum degree at 12 most r has an acyclic (1 + b (r+1) 2 4 c)-coloring. 13

We study weighted bipartite edge coloring problem, which is a generalization of two classical problems: bin packing and edge coloring. This problem has been inspired from the study of Clos networks in multirate switching environment in communication networks. In weighted bipartite edge coloring problem, we are given an edge-weighted bipartite multigraph G = (V,E) with weights w : E → [0, 1]. Th...

The λ-backbone coloring is one of the various problems of vertex colorings in graphs. Given an integer λ ≥ 2, a graph G = (V,E), and a spanning subgraph (backbone) H = (V, EH) of G, a λ-backbone coloring of (G,H) is a proper vertex coloring V → {1, 2, ...} of G in which the colors assigned to adjacent vertices in H differ by at least λ. The λ-backbone coloring number BBCλ(G,H) of (G,H) is the s...