The chromaticity of s-bridge graphs and related graphs

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 and E(G) the edge set of G. Let P(G; A), or simply P(G) if there is no likelihood of confusion, denote the chromatic polynomial of G. In this paper, y=11 and x= -y. Two graphs G and H are said to be chromatically equivalent if P(G) = P(H). A graph G is said to be chromatically unique if P(G) = P(H) implies H is isomorphic to G. Let 93 = {Go, . . . , GP} and x = (Ki,, . . . , Kip >. Then all graphs obtained from Go, . . . , G, by overlapping in Ki, , . . . , Kip in different positions and different orders form a class of graphs. We denote it by ($9, x}. If the chromatic equivalence of H to (9, -X> implies HE{??, X}, then (9,37} is said to be a complete class of chromatically equivalent graphs. For details and other symbols and definitions, readers can see [6]. A 2-connected graph G is called a generalized polygon tree if it can be decomposed into cycle class Y = { Ci,, . . . , Cir} and there exist an overlapping process: H1 = Ci,, Hj is obtained from Hj_1 and Ci, by overlapping in path Pi, where in each step of overlapping, the vertices on Pii, except end vertices, are with degree 2. In [6], it was proved that a 2-connected graph is a generalized polygon tree if and only if it has no subgraphs homeomorphic to K4. Obviously a generalized polygon * Corresponding author. Present address: Rutgers Center for Operations Research, P.O. Box 5062. Rutgers University, New Brunswick, NJ 08903, USA. 0012-365X/94/$07.00 IQ 1994-Elsevier Science B.V. All rights reserved SSDI 0012-365X(93)E0091-H 3.50 S.-J. Xu et al. / Discrete Mathematics 135 (1994) 349-358 tree is a planar graph. For a 2-connected planar graph G, we define r(G) as the number of interior regions of G: r(G)= 1 E(G)1 ( V(G) I+ 1. For a generalized polygon tree, we define intercourse number of G, a(G), as the number of nonadjacent vertex pairs, where there are at least three internally disjoint paths joining them. It was also proved in [6] that if G is a generalized polygon tree and P(H)=P(G), then H is also a generalized polygon tree and r(H) = r(G), o(H) = O(G). A graph consisting of s paths joining two vertices is called an s-bridge graph, which is denoted by F(kI, . . . . k,), where kI, . . . . k, are the lengths of s paths. Clearly an s-bridge graph is a generalized polygon tree. It was proved by Chao and Whitehead [l] that the cycle C, is chromatically unique. Later, Loerinc [3] proved that the generalized O-graph Qobc is chromatically unique. That is to say, 2-bridge graphs and 3-bridge graphs are all chromatically unique. In this paper, we consider the chromaticity of s-bridge graphs and some graphs related to s-bridge graphs. First we give the sufficient and necessary conditions for a 4-bridge graph to be chromatically unique. Now let G be a 4-bridge graph F (a, b,c, d), the lengths of the 4 paths joining two vertices u and v be a, b, c and d, where a> b >c >d and 1 V(G)1 = n. Then IE(G)I=a+b+c+d=n+2. Theorem 1. A 4-bridge graph G is not chromatically unique ifund only ifG sutisjies one of the following conditions: (1) d=l, (2) d=2 and u=b+l=c+2. Proof. If d = 1, G is a polygon tree obtained from C,+ r, Cb+r and Cc+r by overlapping on an edge. Let Q = {C, + r , Cbflr Cc+l >, Xx= {K,,K,}. As shown in C61,{9,X} is a complete class of chromatically equivalent graphs and it is easy to see that l{S,X31>1 (seeFig. l.)H ence G is not chromatically unique and all graphs which are chromatically equivalent to G belong to (9,Xx). In the following, we always assume that d > 1. Then G is a generalized polygon tree, the interior region number r(G)=3 and the intercourse number a(G)= 1. First, using the formula P(G)=P(G+uv)+P(G.uv), we compute the chromatic