Arthur M. Hobbs

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.

The square of a block is Hamiltonian connected

Abstract Let B be a block (finite connected graph without cut-vertices) with at least four vertices and ξ, η be distinct vertices of B . We construct a new block M = M ( B , ξ , η ) containing five copies of B , and use the existence of a Hamiltonian circuit in M 2 to establish the existence of a Hamiltonian path starting at ξ and ending at η in B 2 . A variant of this trick shows that B 2 − ξ has a Hamiltonian circuit.

Hamiltonian Cycles in Regular Graphs

Publisher Summary This chapter presents the theorem of Hamiltonian cycles in regular graphs. If in a graph of order n every vertex has degree at least 1/2 n then the graph contains a Hamiltonian cycle. This theorem is the first in a long line of results concerning forcibly Hamiltonian degree sequences—that is, degree sequences all whose realizations are Hamiltonian. A question is discussed whether a two-connected ( m – k )-regular graph G of order 2 m is Hamiltonian if k (≥1) is sufficiently small. If instead of regularity we ask only that the minimal degree is m – 1 then the answer is negative. The order of k in the example above is best possible: the graph has to be Hamiltonian if k c 1 m – c 2 for some positive constants c 1 , c 2 . It seems very likely that the best value of c 1 is 1/3.

A Class of Hamiltonian Regular Graphs

In this paper, we show that if n 14 and if G is a Z-connected graph with 2n or 2n- 1 vertices which is regular of degree n -2, then G is Hamiltonian if and only if G is not the Petersen graph. We use the terminology of Behzad and Chartrand [2]. In particular, a set of vertices in a graph is independent if no two of the vertices in the set are adjacent. A graph is cubic if every vertex of the graph has degree three. Dirac [6] showed that if G is a graph with m r3 vertices and if every vertex of G has degree irn or more, then G is Hamiltonian. Dirac’s work has been extended in [lo], [ll], [3], [5], [8], and [4], but these results all require the existence of vertices of degree at least irn. Avoiding this latter

Graph family operations

In previous papers, Catlin introduced four functions, denoted S O , S R , S C , and S H , between sets of nite graphs. These functions proved to be very useful in establishing properties of several classes of graphs, including supereulerian graphs and graphs with nowhere zero k-ows for a xed integer k >3. Unfortunately, a subtle error caused several theorems previously published in Catlin (Discrete Math. 160 (1996) 67{80) to be incorrect. In this paper we correct those errors and further explore the relations between these functions, showing that there is a sort of duality between them and that they act as inverses of one another on certain sets of graphs. c 2001 Published by Elsevier Science B.V.

Hamiltonian Cycles in Regular Graphs of Moderate Degree

In this paper we prove that if k is an integer no less than 3, and if G is a two-connected graph with 2n a vertices, a E {0, 1}, which is regular of degree n k, then G is Hamiltonian if a = 0 and n > k2 + k + 1 or if a = I and n > 2k 2 3k =, 3 . We use the notation and terminology of [1] . Gordon [4] has proved that there are only a small number of exceptional graphs with 2n vertices which are not Hamiltonian when all vertices have degree n 1 or more. The present authors proved [3] that if G is a two-connected graph with 2n vertices which is regular of degree n 2 and if n > 6, then G is Hamiltonian. We now partially extend that result to regular graphs of degree n k, k > 3 . Throughout this paper we suppose that G is a graph with 2n a vertices, with a c {0, 1), which is two connected and regular of degree n k, where k is an integer no less than three . Let P be a longest cycle in G, choose a direction around P, let R = V(G) V(P), and let r = I R 1 . For the lemmas, suppose r > 1 . By a theorem of Dirac [2], ((P) , 2n 2k. For v c R, let C„ be the set of vertices of P adjacent to v, let A,; be the set of vertices of P immediately preceding elements of C, in the ordering of P, and let B,, be the set of vertices of P immediately following elements of C,, . The first lemma is trivial .

Disjoint cliques and disjoint maximal independent sets of vertices in graphs

In this paper, we find lower bounds for the maximum and minimum numbers of cliques in maximal sets of pairwise disjoint cliques in a graph. By complementation, these yield lower bounds for the maximum and minimum numbers of independent sets in maximal sets of pairwise disjoint maximal independent sets of vertices in a graph. In the latter context, we show by examples that one of our bounds is best possible.

Maximal Hamiltonian cycles in squares of graphs

In this paper we investigate some of the properties of Hamiltonian cycles in the square G2 of a graph G, which are maximal in that they include as many as possible of the edges of G. We show that there is a maximal Hamiltonian cycle in G2 which has an especially simple structure at each of the cut vertices of G. Further, we show that a maximal Hamiltonian cycle must use all of the edges of any cycle of the graph G which is also a block of G, with “use” being given a precise meaning later. We use the terminology and notation of Behzad and Chartrand [I ] with the few minor exceptions listed here. We denote the degree of a vertex z: in a graph G by deg,(v) and indicate that vertices v and w are adjacent in graph G by v adj, w. The edge joining v and w is denoted by (v, w). We use Q to represent the graph with no vertices and no edges. An edge e of graph G is an end edge if one end of e has degree 1 in G. The square G2 of a graph G is a graph with vertex set V(G) in which two vertices are adjacent if and only if their distance in G is 1 or 2. We denote the number of elements in a set A by / A 1 , and we let C/‘,(G) denote the set of vertices whose degree in G is i. IE is a subgraph of G, then G H = (G E(H)) V,(G E(H)). A tonian cycle h in G2 is maximaI if [ E(h) n E(G)/ > / E(k) n E(G)\ for every Elamiltonian cycle k in G2. Maximal Hamiltonian cycles were introduced in [2] and studied further in [3]. Since paths and cycles are not graphs but sequences, we denote the subgraph whdse vertices and edges are precisely those of a given path or cycle .K: by /k/. With explicit exceptions, all paths and cycles in this paper are represented by sequences of vertices. It is to be understood that an edge of a path or cycle is any edge of the graph which joins vertices which are successive in the path or cycle.

The entire graph of a bridgeless connected plane graph is hamiltonian

Abstract In this paper we show that the entire graph of a bridgeless connected plane graph is hamiltonian, and that the entire graph of a plane block is hamiltonian connected and vertex pancyclic. In addition, we show that in any block G which is not a circuit, given a vertex v of G and a circuit k of G, there is a path p, suspended in G, such that p is a path in k of length at least 1 and G − E(p) − V0(G − E(p)) is a block which includes v.

Balanced and 1-balanced graph constructions

There are several density functions for graphs which have found use in various applications. In this paper, we examine two of them, the first being given by b(G)=|E(G)|/|V(G)|, and the other being given by g(G)=|E(G)|/(|V(G)|-@w(G)), where @w(G) denotes the number of components of G. Graphs for which b(H)@?b(G) for all subgraphs H of G are called balanced graphs, and graphs for which g(H)@?g(G) for all subgraphs H of G are called 1-balanced graphs (also sometimes called strongly balanced or uniformly dense in the literature). Although the functions b and g are very similar, they distinguish classes of graphs sufficiently differently that b(G) is useful in studying random graphs, g(G) has been useful in designing networks with reduced vulnerability to attack and in studying the World Wide Web, and a similar function is useful in the study of rigidity. First we give a new characterization of balanced graphs. Then we introduce a graph construction which generalizes the Cartesian product of graphs to produce what we call a generalized Cartesian product. We show that generalized Cartesian product derived from a tree and 1-balanced graphs are 1-balanced, and we use this to prove that the generalized Cartesian products derived from 1-balanced graphs are 1-balanced.