Multiple Coloring of Cone Graphs

100學年度研究獎補助論文[[abstract]]A k-fold coloring of a graph assigns to each vertex a set of k colors, and color sets assigned to adjacent vertices are disjoint. The kth chromatic number Xk(G) of a graph G is the minimum total number of colors needed in a k-fold coloring of G. Given a graph G = (V, E) and an integer m ≥ 0, the m-cone of G, denoted by µm(G), has vertex set (V x {0,1,… , m}) U {u} in which u is adjacent to every vertex of V x {m}, and (x, i)(y, j) is an edge if xy ∈ E and i = j = 0 or xy ∈ E and |i - j| = 1. This paper studies the kth chromatic number of the cone graphs. An upper bound for Xk(µm(G) in terms of Xk(G), k, and m are given. In particular, it is proved that for any graph G, if m ≥ 2k, then Xk(µm(G)) ≤ Xk(G) + 1. We also find a surprising connection between the kth chromatic number of the cone graph of G and the circular chromatic number of G. It is proved that if Xk(G)/k > Xc((G) and Xk(G) is even, then for sufficiently large m, Xk(µm(G)) = Xk(G). In particular, if X(G) > Xc(G) and X(G) is even, then for sufficiently large m, X(µm(G)) = X(G).[[notice]]補正完畢[[incitationindex]]SCI[[booktype]]紙

The Research on the L(2,1)-labeling problem from Graph theoretic and Graph Algorithmic Approaches

The L(2,1) -labeling problem has been extensively researched on many graph classes. In this thesis, we have also studied the problem on some particular classes of graphs. In Chapter 2 we present a new general approach to derive upper bounds for L(2,1)-labeling numbers and applied that approach to derive bounds for the four standard graph products. In Chapter 3 we study the L(2,1)-labeling number of the composition of n graphs. In Chapter 4 we consider the Cartesian sum of graphs and derive, both, lower and upper bounds for their L(2,1)-labeling number. We use two different approaches to derive the upper bounds and both approaches improve previously known bounds. We also present new approximation algorithms for the L(2,1 )-labeling problem on Cartesian sum graphs. In Chapter 5, we characterize d-disk graphs for d\u3e1, and give the first upper bounds on the L(2,1)-labeling number for this class of graphs. In Chapter 6, we compute upper bounds for the L(2,1)-labeling number of total graphs of K_{1,n}-free graphs. In Chapter 7, we study the four standard products of graphs using the adjacency matrix analysis approach. In Chapter 8, we determine the exact value for the L(2,1)-labeling number of a particular class of Mycielski graphs. We also provide, both, lower and upper bounds for the L(2,1)-labeling number of any Mycielski graph