Smallest maximally nonhamiltonian graphs

A graphG ismaximally nonhamiltonian iffG is not hamiltonian butG + e is hamiltonian for each edgee inG ^c, i.e., any two non-adjacent vertices ofG are ends of a hamiltonian path. Bollobás posed the problem of finding the least number of edges,f(n), possible in a maximally nonhamiltonian graph of ordern. Results of Bondy show thatf(n) 3/2 n forn 7. We exhibit graphs of even ordern 36 for which the bound is attained. These graphs are the snarks,J _k, of Isaacs and mild variations of them. For oddn 55 we construct graphs from the graphsJ _k showing that in this case,f(n) = 3n + 1/2 or 3n + 3/2 and leave the determination of which is correct as an open problem. Finally we note that the graphsJ _k, k 7 are hypohamiltonian cubics with girth 6.     

On translations of double rays in graphs

Let be an infinite graph, let be a double ray in , and letd andd denote the distance functions in and in , respectively. One calls anaxis ifd(x,y)=d (x,y) and aquasi-axis if lim infd(x,y)/d (x,y)>0 asx, y range over the vertex set of andd (x,y). The present paper brings together in greater generality results of R. Halin concerning invariance of double rays under the action of translations (i.e., graph automorphisms all of whose vertex-orbits are infinite) and results of M. E. Watkins concerning existence of axes in locally finite graphs. It is shown that if is a translation whose directionD() is a thin end, then there exists an axis inD() andD(^–1) invariant under ^r for somer not exceeding the maximum number of disjoint rays inD().The thinness ofD() is necessary. Further results give necessary conditions and sufficient conditions for a translation to leave invariant a quasi-axis.     

Pairs of edge disjoint Hamiltonian circuits in 5-connected planar graphs

Summary A variety of examples of 4-connected 4-regular graphs with no pair of disjoint Hamiltonian circuits were constructed in response to Nash-Williams conjecture that every 4-connected 4-regular graph is Hamiltonian and also admits a pair of edge-disjoint Hamiltonian circuits. Nash-Williams's problem is especially interesting for planar graphs since 4-connected planar graphs are Hamiltonian. Examples of 4-connected 4-regular planar graphs in which every pair of Hamiltonian circuits have edges in common are included in the above mentioned examples.B. Grünbaum asked whether 5-connected planar graphs always admit a pair of disjoint Hamiltonian circuits. In this paper we introduce a technique that enables us to construct infinitely many examples of 5-connected planar graphs, 5-regular and non regular, in which every pair of Hamiltonian circuits have edges in common.     

Dirac type condition and Hamiltonian-connected graphs

Abstract

In 1952 Dirac introduced the Dirac type condition and proved that if G is a connected graph of order n ≥ 3 such that δ(G) ≥ n/2, then G is Hamiltonian. In this paper we consider Hamiltonian-connectedness, which extends the Hamiltonian graphs and prove that if G is a connected graph of order n ≥ 3 such that δ(G) ≥ (n ?1)/2, then G is Hamiltonian-connected or G belongs to five families of well-structured graphs. Thus, the condition and the result generalize the above condition and results of Dirac, respectively.     

A note on perfect graphs

Let the lines of a complete graph be 3-colored so that no triangle gets 3 different colors. If two of these colors form perfect graphs then so does the third.     

Heavy cycles in k-connected weighted graphs with large weighted degree sums

A weighted graph is one in which every edge e is assigned a nonnegative number w(e), called the weight of e. The weight of a cycle is defined as the sum of the weights of its edges. The weighted degree of a vertex is the sum of the weights of the edges incident with it. In this paper, we prove that: Let G be a k-connected weighted graph with k?2. Then G contains either a Hamilton cycle or a cycle of weight at least 2m/(k+1), if G satisfies the following conditions: (1) The weighted degree sum of any k+1 pairwise nonadjacent vertices is at least m; (2) In each induced claw and each induced modified claw of G, all edges have the same weight. This generalizes an early result of Enomoto et al. on the existence of heavy cycles in k-connected weighted graphs.     

The partial complement of graphs

For a graphG, the switched graphS _v (G) ofG at a vertexv is the graph obtained fromG by deleting the edges ofG incident withv and adding the edges of incident withv. Properties of graphs whereS _v (G) G or are studied. This concept is extended to the partial complementS _H (G) where H . The investigation here centers around the existence of setsH for whichS _H (G) G. A parameter is introduced which measures how near a graph is to being self-complementary.     

A note on 1-factors in graphs

Conditions on a graphG are presented which are sufficient to guarantee thatG–Z contains a 1-factor, whereZ is a set of edges ofG of restricted cardinality. These conditions provide generalizations of several known results and, further, establish the result that ifG is anr-regular, (r–2)-edge-connected graph (r2) of even order andz is an integer with 0zr–1 such thatG contains fewer thanr–z edge cut sets of cardinalityr–2, thenG–Z has a 1-factor for each setZ ofz edges ofG.     

The edge density of 4-critical planar graphs

Several constructions of 4-critical planar graphs are given. These provide answers to two questions of B. Grünbaum and give improved bounds for the maximum edge density of such graphs.     

2–3 graphs which have Vizing's adjacency property

2–3 graphs in which each vertex is adjacent to at least two vertices of degree 3 are shown to be characterised by the number of vertices of degree 3 adjacent to vertices of degree 3 only.     

Graphs with prescribed degree sets and girth

For a finite nonempty set of integers, each of which is at least two, and an integern 3, the numberf(;n) is defined as the least order of a graph having degree set and girthn. The numberf(;n) is evaluated for several sets and integersn. In particular, it is shown thatf({3, 4}; 5) = 13 andf({3, 4}; 6) = 18.Research of the third author was partially supported by a Faculty Research Fellowship from Western Michigan University.     

Characterizations of connected orthogonality graphs of projections of Rickart ∗-rings

In this paper, we study the orthogonality graphs (see Definition 1.2) of ortholattices. We provide a graph theoretic condition for an ortholattice to be orthomodular. We prove that, the orthogonality graphs of two orthomodular lattices are isomorphic if and only if the lattices are isomorphic. As an application, it is proved that the zero-divisor graph of a Rickart ?-ring is obtained by successively duplicating the vertices of the orthogonality graph of the lattice of projections in the ring. We characterize the finite Rickart ?-rings for which the orthogonality graph of projections is connected.     

Additivity of the crossing number of graphs with connectivity 2

We prove that the crossing number of graphs with connectivity 2 has in certain cases an additive property analogous to that of crossing number of graphs with connectivity ≤1.     

Triples,current graphs and biembeddings

A family of ladder graphs, used by Youngs in his work on the Heawood conjecture, is used to provide constructions of Skolem and related triple systems, triangular biembeddings of certain complete graphs, and genus embeddings of certain complete multipartite graphs.     

On products and line graphs of signed graphs, their eigenvalues and energy

In this article we examine the adjacency and Laplacian matrices and their eigenvalues and energies of the general product (non-complete extended p-sum, or NEPS) of signed graphs. We express the adjacency matrix of the product in terms of the Kronecker matrix product and the eigenvalues and energy of the product in terms of those of the factor graphs. For the Cartesian product we characterize balance and compute expressions for the Laplacian eigenvalues and Laplacian energy. We give exact results for those signed planar, cylindrical and toroidal grids which are Cartesian products of signed paths and cycles.We also treat the eigenvalues and energy of the line graphs of signed graphs, and the Laplacian eigenvalues and Laplacian energy in the regular case, with application to the line graphs of signed grids that are Cartesian products and to the line graphs of all-positive and all-negative complete graphs.     

Large faces in 4-critical planar graphs with minimum degree 4

We prove that the size of the largest face of a 4-critical planar graph with 4 is at most one half the number of its vertices. Letf(n) denote the maximum of the sizes of largest faces of all such graphs withn vertices (n sufficiently large). We present an infinite family of graphs that shows .All three authors gratefully acknowledge the support of the National Science and Engineering Research Council of Canada.     

Constructing cospectral graphs

Some new constructions for families of cospectral graphs are derived, and some old ones are considerably generalized. One of our new constructions is sufficiently powerful to produce an estimated 72% of the 51039 graphs on 9 vertices which do not have unique spectrum. In fact, the number of graphs of ordern without unique spectrum is believed to be at leastn ³ g _–1 for some>0, whereg _n is the number of graphs of ordern andn 7.     

Connected graphs containing a given connected graph as a unique greatest common subgraph

Those connected graphsG are determined for which there exist nonisomorphic connected graphs of equal size containingG as a unique greatest common subgraph. Analogous results are also obtained for weakly connected and strongly connected digraphs, as well as for induced subgraphs and induced subdigraphs.This research was supported by a Western Michigan University faculty research fellowship.This research was supported in part by a Western Michigan University research assistantship from the Graduate College and the College of Arts and Sciences.