Chromatic equivalence class of the join of certain tripartite graphs

For a simple graph G, let P(G;λ) be the chromatic polynomial of G. Two graphs G and H are said to be chromatically equivalent, denoted G ~ H if P(G;λ) = P(H;λ). A graph G is said to be chromatically unique, if H ~ G implies that H ≅ G. Chia [4] determined the chromatic equivalence class of the graph consisting of the join of p copies of the path each of length 3. In this paper, we determined the chromatic equivalence class of the graph consisting of the join of p copies of the complete tripartite graph K1,2,3. MSC: 05C15;05C6

Chromatic equivalence classes of certain generalized polygon trees, III

AbstractLet P(G) denote the chromatic polynomial of a graph G. Two graphs G and H are chromatically equivalent, if P(G)=P(H). A set of graphs S is called a chromatic equivalence class if for any graph H that is chromatically equivalent with a graph G in S, then H∈S. Peng et al. (Discrete Math. 172 (1997) 103–114), studied the chromatic equivalence classes of certain generalized polygon trees. In this paper, we continue that study and present a solution to Problem 2 in Koh and Teo (Discrete Math. 172 (1997) 59–78)

Chromatic Equivalence Classes and Chromatic Defining Numbers of Certain Graphs

There are two parts in this dissertation: the chromatic equivalence classes and the chromatic defining numbers of graphs. In the first part the chromaticity of the family of generalized polygon trees with intercourse number two, denoted by Cr (a, b; c, d), is studied. It is known that Cr( a, b; c, d) is a chromatic equivalence class if min {a, b, c, d} ≥ r+3. We consider Cr( a, b; c, d) when min{ a, b, c, d} ≤ r + 2. The necessary and sufficient conditions for Cr(a, b; c, d) with min {a, b, c, d} ≤ r + 2 to be a chromatic equivalence class are given. Thus, the chromaticity of Cr (a, b; c, d) is completely characterized. In the second part the defining numbers of regular graphs are studied. Let d(n, r, X = k) be the smallest value of defining numbers of all r-regular graphs of order n and the chromatic number equals to k. It is proved that for a given integer k and each r ≥ 2(k - 1) and n ≥ 2k, d(n, r, X = k) = k - 1. Next, a new lower bound for the defining numbers of r-regular k-chromatic graphs with k < r < 2( k - 1) is found. Finally, the value of d( n , r, X = k) when k < r < 2(k - 1) for certain values of n and r is determined

Chromatic equivalence classes of complete tripartite graphs

AbstractSome necessary conditions on a graph which has the same chromatic polynomial as the complete tripartite graph Km,n,r are developed. Using these, we obtain the chromatic equivalence classes for Km,n,n (where 1≤m≤n) and Km1,m2,m3 (where |mi−mj|≤3). In particular, it is shown that (i) Km,n,n (where 2≤m≤n) and (ii) Km1,m2,m3 (where |mi−mj|≤3, 2≤mi,i=1,2,3) are uniquely determined by their chromatic polynomials. The result (i), proved earlier by Liu et al. [R.Y. Liu, H.X. Zhao, C.Y. Ye, A complete solution to a conjecture on chromatic uniqueness of complete tripartite graphs, Discrete Math. 289 (2004) 175–179], answers a conjecture (raised in [G.L. Chia, B.H. Goh, K.M. Koh, The chromaticity of some families of complete tripartite graphs (In Honour of Prof. Roberto W. Frucht), Sci. Ser. A (1988) 27–37 (special issue)]) in the affirmative, while result (ii) extends a result of Zou [H.W. Zou, On the chromatic uniqueness of complete tripartite graphs Kn1,n2,n3 J. Systems Sci. Math. Sci. 20 (2000) 181–186]

On Minimum Average Stretch Spanning Trees in Polygonal 2-trees

A spanning tree of an unweighted graph is a minimum average stretch spanning tree if it minimizes the ratio of sum of the distances in the tree between the end vertices of the graph edges and the number of graph edges. We consider the problem of computing a minimum average stretch spanning tree in polygonal 2-trees, a super class of 2-connected outerplanar graphs. For a polygonal 2-tree on $n$ vertices, we present an algorithm to compute a minimum average stretch spanning tree in $O(n \log n)$ time. This algorithm also finds a minimum fundamental cycle basis in polygonal 2-trees.Comment: 17 pages, 12 figure

Chromaticity Of Certain K4-Homeomorphs

The chromaticity of graphs is the term used referring to the question of chromatic equivalence and chromatic uniqueness of graphs. Since the arousal of the interest on the chromatically equivalent and chromatically unique graphs, various concepts and results under the said areas of research have been discovered and many families of such graphs have been obtained. The purpose of this thesis is to contribute new results on the chromatic equivalence and chromatic uniqueness of graphs, specifically, K4-homeomorphs

Chromatic equivalence classes of certain generalized polygon trees

Let P(G) denote the chromatic polynomial of a graph G. Two graphs G and H are chromatically equivalent, written G ∼ H, if P(G) = P(H). Let g denote the family of all generalized polygon trees with three interior regions. Xu (1994) showed that g is a union of chromatic equivalence classes under the equivalence relation '∼'. In this paper, we determine infinitely many chromatic equivalence classes in g under '∼'. As a byproduct, we obtain a family of chromatically unique graphs established by Peng (1995)

Chromaticity of a family of K4 homeomorphs

AbstractA K4 homeomorph can be described as a graph on n vertices having 4 vertices of degree 3 and n − 4 vertices of degree 2; each pair of degree 3 vertices is joined by a path. We study the chromatic uniqueness and chromatic equivalence of one family of K4 homeomorphs. This family has exactly 3 paths of length one. The results of this study leads us to solve 3 of the problems posed by Koh and Teo in their 1990 survey paper which appeared in Graphs and Combinatorics