Hong-Jian Lai

Group Connectivity of 3-Edge-Connected Chordal Graphs

Abstract. Let A be a finite abelian group and G be a digraph. The boundary of a function f: E(G)↦A is a function ∂f: V(G)↦A given by ∂f(v)=∑e leaving vf(e)−∑e entering vf(e). The graph G is A-connected if for every b: V(G)↦A with ∑v∈ V(G)b(v)=0, there is a function f: E(G)↦A{0} such that ∂f=b. In [J. Combinatorial Theory, Ser. B 56 (1992) 165–182], Jaeger et al showed that every 3-edge-connected graph is A-connected, for every abelian group A with |A|≥6. It is conjectured that every 3-edge-connected graph is A-connected, for every abelian group A with |A|≥5; and that every 5-edge-connected graph is A-connected, for every abelian group A with |A|≥3.¶ In this note, we investigate the group connectivity of 3-edge-connected chordal graphs and characterize 3-edge-connected chordal graphs that are A-connected for every finite abelian group A with |A|≥3.

Graphs without spanning closed trails

Jaeger [J. Graph Theory 3 (1979) 91-93] proved that if a graph has two edge-disjoint spanning trees, then it is supereulerian, i.e., that it has a spanning closed trail. Catlin [J. Graph Theory 12 (1988) 29-45] showed that if G is one edge short of having two edge-disjoint spanning trees, then G has a cut edge or G is supereulerian. Catlin conjectured that if a connected graph G is at most two edges short of having two edge-disjoint spanning trees, then either G is supereulerian or G can be contracted to a K2 or a K2,t for some odd integer t 1. We prove Catlin’s conjecture in a more general context. Applications to spanning trails are discussed.

Matrices in Combinatorics and Graph Theory

Combinatorics and Matrix Theory have a symbiotic, or mutually beneficial, relationship. This relationship is discussed in my paper The symbiotic relationship of combinatorics and matrix theoryl where I attempted to justify this description. One could say that a more detailed justification was given in my book with H. J. Ryser entitled Combinatorial Matrix Theon? where an attempt was made to give a broad picture of the use of combinatorial ideas in matrix theory and the use of matrix theory in proving theorems which, at least on the surface, are combinatorial in nature. In the book by Liu and Lai, this picture is enlarged and expanded to include recent developments and contributions of Chinese mathematicians, many of which have not been readily available to those of us who are unfamiliar with Chinese journals. Necessarily, there is some overlap with the book Combinatorial Matrix Theory. Some of the additional topics include: spectra of graphs, eulerian graph problems, Shannon capacity, generalized inverses of Boolean matrices, matrix rearrangements, and matrix completions. A topic to which many Chinese mathematicians have made substantial contributions is the combinatorial analysis of powers of nonnegative matrices, and a large chapter is devoted to this topic. This book should be a valuable resource for mathematicians working in the area of combinatorial matrix theory. Richard A. Brualdi University of Wisconsin - Madison 1 Linear Alg. Applies., vols. 162-4, 1992, 65-105 2Camhridge University Press, 1991.

Fractional arboricity, strength, and principal partitions in graphs and matroids

Abstract In a 1983 paper, D. Gusfield introduced a function which is called (following W.H. Cunningham, 1985) the strength of a graph or matroid. In terms of a graph G with edge set E(G) and at least one link, this is the function η(G) = minF⊆E(G) ∣F∣/(ω(G − F) − ω(G)), where the minimum is taken over all subsets F of E(G) such that ω(G − F), the number of components of G − F, is at least ω(G) + 1. In a 1986 paper, C. Payan introduced the fractional arboricity of a graph or matroid. In terms of a graph G with edge set E(G) and at least one link this function is γ(G) = maxH⊆G ∣E(H)∣/(∣V(H)∣ − ω(H)), where H runs over all subgraphs of G having at least one link. Connected graphs G for which γ(G) = η(G) were used by A. Rucinski and A. Vince in 1986 while studying random graphs. We characterize the graphs and matroids G for which γ(G) = η(G). The values of γ and η are computed for certain graphs, and a recent result of Erdos (that if each edge of G lies in a C3, then ∣E(G)∣≥ 3 2 (∣V(G)∣ − 1)) is generalized in terms of η. The principal partition of a graph was introduced in 1967 by G. Kishi and Y. Kajitani, by T. Ohtsuki, Y. Ishizaki, and H. Watanabe, and by M. Iri (all of these were published in 1968). It has been used since then for the analysis of electrical networks in which the two Kirchhoff laws and Ohms law hold, because it often allows the currents and voltage drops in the network to be completely computed with fewer measurements than are required for either of the Kirchhoff laws used alone. J. Bruno and L. Weinberg generalized the principal partition to matroids in 1971, and their generalization was refined independently by N. Tomizawa (1976) and by H. Narayanan and M.N. Vartak (1974, 1981). Here we demonstrate that γ and η are closely related to the principal partition and can be used to give a simple definition of both the principal partition and the more recent refinements of it.

Eulerian subgraphs and Hamilton-connected line graphs

Let C(l,k) denote a class of 2-edge-connected graphs of order n such that a graph G ∈ C(l,k) if and only if for every edge cut S ⊆ E(G) with |S| ≤ 3, each component of G-S has order at least (n-k)/l. We prove the following: (1) If G ∈ C(6,0), then G is supereulerian if and only if G cannot be contracted to K2,3, K2,5 or K2,3(e), where e ∈ E(K2,3) and K2,3(e) stands for a graph obtained from K2,3 by replacing e by a path of length 2. (2) If G ∈ C(6, 0) and n≥7, then L(G) is Hamilton-connected if and only if κ(L(G))≥3. Former results by Catlin and Li, and by Broersma and Xiong are extended.

Hamiltonian-connected graphs

For a simple graph G, let NCD(G)=min{|N(u)@?N(v)|+d(w):u,v,w@?V(G),uv@?@?E(G),wvorwu@?@?E(G)}. In this paper, we prove that if NCD(G)>=|V(G)|, then either G is Hamiltonian-connected, or G belongs to a well-characterized class of graphs. The former results by Dirac, Ore and Faudree et al. are extended.

Supereulerian Graphs and the Petersen Graph

Any 3-edge-connected graph with at most 10 edge cuts of size 3 either has a spanning closed trail or it is contractible to the Petersen graph.

On the Hamiltonian index

Abstract For simple connected graphs that are neither paths nor cycles, we define h ( G ) = min{ m : L m ( G ) is Hamiltonian} and l ( G ) = max{ m : G has an arc of length m that is not both of length 2 and in a K 3 }, where an arc in G is a path in G whose internal vertices have degree two in G . We prove that h ( G )⩽ l ( G ) + 1. As consequences, we obtain theorems of Chartrand and Wall and of Lesniak-Foster and Williamson. We also characterize those graphs that satisfy l ( G ) + 1 = h ( G ). This characterization provides counterexamples to a previous result in [5].

Group Chromatic Number of Graphs without K 5 -Minors

Abstract. Let G be a graph with a fixed orientation and let A be a group. Let F(G,A) denote the set of all functions f: E(G) ↦A. The graph G is A-colorable if for any function f∈F(G,A), there is a function c: V(G) ↦A such that for every directed e=uv∈E(G), c(u)−c(v)≠f(e). The group chromatic numberχ1(G) of a graph G is the minimum m such that G is A-colorable for any group A of order at least m under a given orientation D.In [J. Combin. Theory Ser. B, 56 (1992), 165–182], Jaeger et al. proved that if G is a simple planar graph, then χ1(G)≤6. We prove in this paper that if G is a simple graph without a K5-minor, then χ1(G)≤5.

Graph whose edges are in small cycles

Abstract Paulraja (1987) conjectured the following: 1. (i) If every edge of a 2-connected graph G lies in a cycle of length at most 4 in G , then G has a dominating closed trail. 2. (ii) If, in addition, δ( G )⩾3, then G has a closed spanning trail. Collapsible graphs are defined and studied by Catlin (1988). Catlin showed that if H is a collapsable subgraph of G , then G has a spanning closed trail if and only if G/H , the graph obtained from G by contracting H , has a spanning closed trail. Catlin (1987) conjectured that a graph satisfying the hypothesis of (ii) is collapsable. In this paper, all three conjectures are proved.