Optimal Edge-Colourings for a Class of Planar Multigraphs

  Let G be a multigraph containing no minor isomorphic to or (where denotes without one of its edges). We show that the chromatic index of G is given by , where is the maximum valency of G and is defined as

Entropy bounds for perfect matchings and Hamiltonian cycles

For a graph G = (V,E) and x: E → ℜ⁺ satisfying Σ _e∋υ x _e = 1 for each υ ∈ V, set h(x) = Σ _e x _e log(1/x _e ) (with log = log₂). We show that for any n-vertex G, random (not necessarily uniform) perfect matching f satisfying a mild technical condition, and x _e =Pr(e∈f),

Induced Ramsey Numbers

   We investigate the induced Ramsey number of pairs of graphs (G, H). This number is defined to be the smallest possible order of a graph Γ with the property that, whenever its edges are coloured red and blue, either a red induced copy of G arises or else a blue induced copy of H arises. We show that, for any G and H with , we have

On the full automorphism group of a graph

While it is easy to characterize the graphs on which a given transitive permutation groupG acts, it is very difficult to characterize the graphsX with Aut (X)=G. We prove here that for the certain transitive permutation groups a simple necessary condition is also sufficient. As a corollary we find that, whenG is ap-group with no homomorphism ontoZ _p wrZ _p , almost all Cayley graphs ofG have automorphism group isomorphic toG.     

Random Graph Coverings I: General Theory and Graph Connectivity

In this paper we describe a simple model for random graphs that have an n-fold covering map onto a fixed finite base graph. Roughly, given a base graph G and an integer n, we form a random graph by replacing each vertex of G by a set of n vertices, and joining these sets by random matchings whenever the corresponding vertices are adjacent in G. The resulting graph covers the original graph in the sense that the two are locally isomorphic. We suggest possible applications of the model, such as constructing graphs with extremal properties in a more controlled fashion than offered by the standard random models, and also "randomizing" given graphs. The main specific result that we prove here (Theorem 1) is that if is the smallest vertex degree in G, then almost all n-covers of G are -connected. In subsequent papers we will address other graph properties, such as girth, expansion and chromatic number. Received June 21, 1999/Revised November 16, 2000 RID="*" ID="*" Work supported in part by grants from the Israel Academy of Aciences and the Binational Israel-US Science Foundation.     

Drawing 4-Pfaffian graphs on the torus

We say that a graph G is k-Pfaffian if the generating function of its perfect matchings can be expressed as a linear combination of Pfaffians of k matrices corresponding to orientations of G. We prove that 3-Pfaffian graphs are 1-Pfaffian, 5-Pfaffian graphs are 4-Pfaffian and that a graph is 4-Pfaffian if and only if it can be drawn on the torus (possibly with crossings) so that every perfect matching intersects itself an even number of times. We state conjectures and prove partial results for k>5. The author was supported in part by NSF under Grant No. DMS-0200595 and DMS-0701033.     

Tutte sets in graphs II: The complexity of finding maximum Tutte sets

A well-known formula of Tutte and Berge expresses the size of a maximum matching in a graph G in terms of what is usually called the deficiency. A subset X of V(G) for which this deficiency is attained is called a Tutte set of G. While much is known about maximum matchings, less is known about the structure of Tutte sets. We explored the structural aspects of Tutte sets in another paper. Here, we consider the algorithmic complexity of finding Tutte sets in a graph. We first give two polynomial algorithms for finding a maximal Tutte set. We then consider the complexity of finding a maximum Tutte set, and show it is NP-hard for general graphs, as well as for several interesting restricted classes such as planar graphs. By contrast, we show we can find maximum Tutte sets in polynomial time for graphs of level 0 or 1, elementary graphs, and 1-tough graphs.     

Disproof of a conjecture in graph reconstruction theory

In his thesis [3] B. D. Thatte conjectured that ifG=G ₁,G ₂,...G _n is a sequence of finitely many simple connected graphs (isomorphic graphs may occur in the sequence) with the same number of vertices and edges then their shuffled edge deck uniquely determines the graph sequence (up to a permutation). In this paper we prove that there are such sequences of graphs with the same shuffled edge deck.This research was partially supported by Hungarian National Foundation of Scientific Research Grant no. 1812     

Alternating cycle-free matchings

The investigation of alternating cycle-free matchings is motivated by the Jump-number problem for partially ordered sets and the problem of counting maximum cardinality matchings in hexagonal systems.We show that the problem of deciding whether a given chordal bipartite graph has an alternating cycle-free matching of a given cardinality is NP-complete. A weaker result, for bipartite graphs only, has been known for some time. Also, the alternating cycle-free matching problem remains NP-complete for strongly chordal split graphs of diameter 2.In contrast, we give algorithms to solve the alternating cycle-free matching problem in polynomial time for bipartite distance hereditary graphs (time O(m ²) on graphs with m edges) and distance hereditary graphs (time O(m ⁵)).     

Random Matchings in Regular Graphs

d -regular graph G, let M be chosen uniformly at random from the set of all matchings of G, and for let be the probability that M does not cover x. We show that for large d, the 's and the mean μ and variance of are determined to within small tolerances just by d and (in the case of μ and ) : Theorem. For any d-regular graph G, (a) , so that , (b) , where the rates of convergence depend only on d. Received: April 12, 1996     

Ordering n-vertex cacti with matching number q by their spectral radii

Abstract

A connected graph G is a cactus if any two of its cycles have at most one common vertex. Denote by the set of n-vertex cacti with matching number q. Huang, Deng and Simi? [23] identified the unique graph with the maximum spectral radius among 2q-vertex cacti with perfect matchings. In this paper, as a continuance of it, the largest and second largest spectral radii together with the corresponding graphs among are determined. Consequently, the first two largest spectral radii together with cacti having perfect matchings are also determined.     

Graph colourings with forbidden k-coloured subgraphs

Abstract

For a set F of graphs and a natural number k, an (F, k)-colouring of a graph G is a proper colouring of V (G) such that no subgraph of G isomorphic to an element of F is coloured with at most k colours. Equivalently, if P is the class of all graphs that do not contain an element of F as a subgraph, a χP,k colouring of G is a proper colouring such that the union of at most k colour classes induces a graph in P. The smallest number of colours in such a colouring of G, if it exists, is denoted by χP,k (G). We give some general results on χP,k-colourings and investigate values of χP,k (G) for some choices of P and classes of graphs G.     

Packing random rectangles

A random rectangle is the product of two independent random intervals, each being the interval between two random points drawn independently and uniformly from [0,1]. We prove that te number C _n of items in a maximum cardinality disjoint subset of n random rectangles satisfies

Maximal Energy Bipartite Graphs

 Given a graph G, its energy E(G) is defined to be the sum of the absolute values of the eigenvalues of G. This quantity is used in chemistry to approximate the total π-electron energy of molecules and in particular, in case G is bipartite, alternant hydrocarbons. Here we show that if G is a bipartite graph with n vertices, then

Altitude of regular graphs with girth at least five

The altitude of a graph G is the largest integer k such that for each linear ordering f of its edges, G has a (simple) path P of length k for which f increases along the edge sequence of P. We determine a necessary and sufficient condition for cubic graphs with girth at least five to have altitude three and show that for r?4, r-regular graphs with girth at least five have altitude at least four. Using this result we show that some snarks, including all but one of the Blanus?a type snarks, have altitude three while others, including the flower snarks, have altitude four. We construct an infinite class of 4-regular graphs with altitude four.     

On the null-spaces of acyclic and unicyclic singular graphs

For acyclic and unicyclic graphs we determine a necessary and sufficient condition for a graph G to be singular. Further, it is shown that this characterization can be used to construct a basis for the null-space of G.     

Counting perfect matchings in n-extendable graphs

The structural theory of matchings is used to establish lower bounds on the number of perfect matchings in n-extendable graphs. It is shown that any such graph on p vertices and q edges contains at least ⌈(n+1)!/4[q-p-(n-1)(2Δ-3)+4]⌉ different perfect matchings, where Δ is the maximum degree of a vertex in G.     

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.     

Graphs with independent perfect matchings

A graph with at least two vertices is matching covered if it is connected and each edge lies in some perfect matching. A matching covered graph G is extremal if the number of perfect matchings of G is equal to the dimension of the lattice spanned by the set of incidence vectors of perfect matchings of G. We first establish several basic properties of extremal matching covered graphs. In particular, we show that every extremal brick may be obtained by splicing graphs whose underlying simple graphs are odd wheels. Then, using the main theorem proved in 2 and 3 , we find all the extremal cubic matching covered graphs. © 2004 Wiley Periodicals, Inc. J Graph Theory 48: 19–50, 2005     

Some properties of the spectrum of graphs

Let G be a graph and denote by Q(G)=D(G) A(G),L(G)=D(G)-A(G) the sum and the difference between the diagonal matrix of vertex degrees and the adjacency matrix of G,respectively. In this paper,some properties of the matrix Q(G)are studied. At the same time,anecessary and sufficient condition for the equality of the spectrum of Q(G) and L(G) is given.