A multi-bridge hypergraph is an h-uniform linear hypergraph consisting of some linear paths having common extremities. In this paper it is proved that the multisets of path lengths of two chromatically equivalent multi-bridge hypergraphs are equal provided the multiplicities of path lengths are bounded above by 2h−1 − 2. ∗This research is partially supported by Abdus Salam School of Mathematica...

1.1 In this note we compute the chromatic polynomial of the Jahangir graph J2p and we prove that it is chromatically unique for p = 3. AMS Subject classification: 05C15

In this note it is shown that every hypergraph containing a pendant path of length at least 2 is not chromatically unique. The same conclusion holds for h-uniform r-quasi linear 3-cycle if r ≥ 2.

The graph consisting of s paths joining two vertices is called an s-bridge graph. In this paper, we discuss the chromaticity of some families of s-bridge graphs, especially 4-bridge graphs, and some graphs related to s-bridge graphs. 1. The chromaticity of 4-bridge graphs The graphs considered here are finite, undirected, simple and loopless. For a graph G, let V(G) denote the vertex set of G a...

Let \(P(G, x)\) be a chromatic polynomial of graph \(G\). Two graphs \(G\) and \(H\) are called chromatically equivalent iff x) = H(G, x)\). A is unique if \(G\simeq H\) for every to In this paper, the uniqueness complete tripartite \(K(n_1, n_2, n_3)\) proved \(n_1 \geqslant n_2 n_3 2\) - \leqslant 5\).

Let P(G;) denote the chromatic polynomial of a graph G. A graph G is chromatically unique if G ∼ = H for any graph H such that P(H;) = P(G;). This note corrects an error in the proof of the chromatic uniqueness of certain 2-connected graphs with n vertices and n + 3 edges.

In this note, all chromatic equivalence classes for 2-connected 3-chromatic graphs with five triangles and cyclomatic number six are described. New families of chromatically unique graphs of order n are presented for each n ≥ 8. This is a generalization of a result stated in [5]. Moreover, a proof for the conjecture posed in [5] is given.

A K4-homeomorph is a subdivision of the complete graph with four vertices (K4). Such a homeomorph is denoted by K4(a,b,c,d,e,f) if the six edges of K4 are replaced by the six paths of length a,b,c,d,e,f, respectively. In this paper, we discuss the chromaticity of a family of K4-homeomorphs with girth 10. We also give sufficient and necessary condition for some graphs in the family to be chromat...

This paper initiates a study of the connection between graph homomorphisms and the Tutte polynomial. This connection enables us to extend the study to other important polynomial invariants associated with graphs, and closely related to the Tutte polynomial. We then obtain applications of these relationships in several areas, including Abelian Groups and Statistical Physics. A new type of unique...