Abstract: It is proved that the hamiltonian index of a connected graph other than a path is less than its diameter which improves the results of P. A. Catlin etc. [J. Graph Theory 14 (1990) 347–364] and M. L. Sarazin [Discrete Math. 134(1994)85–91]. Nordhaus-Gaddum's inequalities for the hamiltonian index of a graph are also established.

Abstract: An s-arc in a graph is a vertex sequence (α0,α1,…,αs) such that {αi−1,αi} ∈ EΓ for 1 [les ] i [les ] s and αi−1 ≠ αi+1 for 1 [les ] i [les ] s − 1. This paper gives a characterization of a class of s-transitive graphs; that is, graphs for which the automorphism group is transitive on s-arcs but not on (s + 1)-arcs. It is proved that if Γ is a finite connected s-transitive graph (where s [ges ] 2) of order a p-power with p prime, then s = 2 or 3; further, either s = 3 and Γ is a normal cover of the complete bipartite graph K2m,2m, or s = 2 and Γ is a normal cover of one of the following 2-transitive graphs: Kpm+1 (the complete graph of order pm+1), K2m,2m − 2mK2 (the complete bipartite graph of order 2m+1 minus a 1-factor), a primitive affine graph, or a biprimitive affine graph. (Finite primitive and biprimitive affine 2-arc transitive graphs were classified by Ivanov and Praeger in 1993.)

Abstract: In this chapter we return to the theme of combinatorial regularity with the study of strongly regular graphs. In addition to being regular, a strongly regular graph has the property that the number of common neighbours of two distinct vertices depends only on whether they are adjacent or nonadjacent. A connected strongly regular graph with connected complement is just a distance-regular graph of diameter two. Any vertex-transitive graph with a rank-three automorphism group is strongly regular, and we have already met several such graphs, including the Petersen graph, the Hoffman-Singleton graph, and the symplectic graphs of Section 8.11.

Abstract: We study the totally transitive graph maps which have periodic points. In particular we show that the cofiniteness of the set of periods characterizes these maps among transitive graph maps.

Abstract: It is shown that every 4-chromatic graph on n vertices contains an odd cycle of length less than $2\sqrt {n}\,+3$. This improves the previous bound given by Nilli [J Graph Theory 3 (1999), 145–147]. © 2001 John Wiley & Sons, Inc. J Graph Theory 37: 115–117, 2001 1990 Mathematics Subject Classification 05C35, 05C55.

Abstract: A (1, 2)-eulerian weight w of a cubic graph is called a Hamilton weight if every faithful circuit cover of the graph with respect to w is a set of two Hamilton circuits. Let G be a 3-connected cubic graph containing no Petersen minor. It is proved in this paper that G admits a Hamilton weight if and only if G can be obtained from K4 by a series of Δ↔Y-operations. As a byproduct of the proof of the main theorem, we also prove that if G is a permutation graph and w is a (1,2)-eulerian weight of G such that (G, w) is a critical contra pair, then the Petersen minor appears “almost everywhere” in the graph G. © 2001 John Wiley & Sons, Inc. J Graph Theory 38: 197–219, 2001 Dedicated to Professor Yong-Jin Zhu for his 70th Birthday.

Abstract: This note contains an example of a 4-chromatic graph which admits a vertex partition into three parts such that the union of every two of them induces a forest. © 2001 John Wiley & Sons, Inc. J Graph Theory 37: 243–246, 2001

Abstract: Let $G$ be an undirected and loopless finite graph that is not a path. The minimum $m$ such that the iterated line graph $L^m(G)$ is hamiltonian is called the hamiltonian index of $G,$ denoted by $h(G).$ A reduction method to determine the hamiltonian index of a graph $G$ with $h(G)\geq 2$ is given here. With it we will establish a sharp lower bound and a sharp upper bound for $h(G)$, respectively, which improves some known results of P.A. Catlin et al. [J. Graph Theory 14 (1990)] and H.-J. Lai [Discrete Mathematics 69 (1988)]. Examples show that $h(G)$ may reach all integers between the lower bound and the upper bound.

Abstract: In this paper, it is shown that the graph obtained by overlapping the cycle and the complete tripartite graph at an edge is uniquely determined by its chromatic polynomial. ) 3 ( ≥ m C m 2 , 2 , 2 K r r r K "

Abstract: An arc in a graph is an ordered pair of adjacent vertices, and so a graph is arc-transitive if its automorphism group acts transitively on the set of arcs. As we have seen, this is a stronger property than being either vertex transitive or edge transitive, and so we can say even more about arc-transitive graphs. The first few sections of this chapter consider the basic theory leading up to Tutte’s remarkable results on cubic arc-transitive graphs. We then consider some examples of arc-transitive graphs, including three of the most famous graphs of all: the Petersen graph, the Coxeter graph, and Tutte’s 8-cage.

Abstract: The concept of the incidence chromatic number of a graph was introduced by Brualdi and Massey They conjectured that every graph G can be incidence colored with Δ (G) +2 colors In this paper, the trueness of this conjecture for complete k partite graph was proved, and the incidence chromatic number of complete k partite graphs was calculated

Abstract: For two integers k(> 0) and s(≥ 0), a cycle of length l is called an (s mod k)-cycle if l ≡ s mod k. In this paper, the following conjecture of Chen, Dean, and Shreve [5] is proved: Every 2-connected graph with at least six vertices and minimum degree at least three contains a (2 mod 4)-cycle.

Abstract: A heuristic for the visualization of arbitrary automorphisms of a graph by two-dimensional drawings is presented. The restriction of the drawing to a subgraph induced by an orbit of the automorphism is according to a symmetry of the plane. For a vertex-symmetric graph, a collection of drawings for a set of automorphisms which generate a transitive group on the vertices shows this symmetry property.

Abstract: An orientation of an undirected graph G is a directed graph D obtained by giving an orientation to every edges of G. An orientation D is complete iff whenever there are edges ab, bc, there is an edge ac in D. A graph G is transitive iff G has a complete orientation. If there are edges ab, bc and an anti-path (a,c) (i.e., a path in the complement of G) in the subgraph induced by the neighbors of b in G, the triplet ( a,b,c) is called a pivot on G and the middle vertex b is the pivot-vertex of it. We proved in T14 that a graph G is transitive iff G has no pivot-cycles (i.e., a closed odd sequence of pivot-vertices) and got a polynomialtime algorithm to recognize a transitive graph and to construct a complete orientation of the graph.

Abstract: A graph with diameter d has girth at most 2d + 1, while a bipartite graph with diameter d has girth at most 2d. While these are very simple bounds, the graphs that arise when they are met are particularly interesting. Graphs with diameter d and girth 2d + 1 are known as Moore graphs. They were introduced by Hoffman and Singleton in a paper that can be viewed as one of the prime sources of algebraic graph theory. After considerable development, the tools they used in this paper led to a proof that a Moore graph has diameter at most two. They themselves proved that a Moore graph of diameter two must be regular, with valency 2, 3, 7, or 57. We will provide the machinery to prove this last result in our work on strongly regular graphs in Chapter 10.

Abstract: We prove that the copnumber of a finite connected graph of genus g is bounded by [3/2g]+3. In particular this means that the copnumber of a toroidal graph is bounded by 4. We also sketch a proof that the copnumber of a graph of genus 2 is bounded by 5.