Unique Enresdowed Graphs

Chromatically Unique Multibridge Graphs

Let θ(a1, a2, · · · , ak) denote the graph obtained by connecting two distinct vertices with k independent paths of lengths a1, a2, · · · , ak respectively. Assume that 2 ≤ a1 ≤ a2 ≤ · · · ≤ ak. We prove that the graph θ(a1, a2, · · · , ak) is chromatically unique if ak < a1 + a2, and find examples showing that θ(a1, a2, · · · , ak) may not be chromatically unique if ak = a1 + a2.

Chromati ally Unique Multibridge Graphs

Classes of chromatically unique graphs

Borowiecki, M. and E. Drgas-Burchardt, Classes of chromatically unique graphs, Discrete Mathematics Ill (1993) 71-75. We prove that graphs obtained from complete equibipartite graphs by deleting some independent sets of edges are chromatically unique. 1. Preliminary definitions and results In this paper we consider finite, undirected, simple and loopless graphs. Two graphs G and H are said to b...

On Unique Independence Weighted Graphs

An independent set in a graph G is a set of vertices no two of which are joined by an edge. A vertex-weighted graph associates a weight with every vertex in the graph. A vertex-weighted graph G is called a unique independence vertex-weighted graph if it has a unique independent set with maximum sum of weights. Although, in this paper we observe that the problem of recognizing unique independenc...

Perfectly Orderable Graphs and Unique Colorability

Given a linear order < on the vertices of a graph, an obstruction is an induced P4 abcd such that a < b and d < c. A linear order without any obstruction is called perfect. A graph is perfectly orderable if its vertex set has some perfect order. In the graph G, for two vertices x and y, x clique-dominates y if every maximum size clique containing y, contains x too. We prove the following result...