Robert E. L. Aldred

Vertex colouring edge partitions

A partition of the edges of a graph G into sets {S1,..., Sk} defines a multiset Xv for each vertex v where the multiplicity of i in Xv is the number of edges incident to v in Si We show that the edges of every graph can be partitioned into 4 sets such that the resultant multisets give a vertex colouring of G. In other words, for every edge (u, v) of G, Xu ≠ Xv. Furthermore, if G has minimum degree at least 1000, then there is a partition of E(G) into 3 sets such that the corresponding multisets yield a vertex colouring.

Algorithms for Pattern Involvement in Permutations

We consider the problem of developing algorithms for the recognition of a fixed pattern within a permutation. These methods are based upon using a carefully chosen chain or tree of subpatterns to build up the entire pattern. Generally, large improvements over brute force search can be obtained. Even using on-line versions of these methods allow for such improvements, though often not as great as for the full method. Furthermore, by using carefully chosen data structures to fine tune the methods, we establish that any pattern of length 4 can be detected in O(n log n) time. We also improve the complexity bound for detection of a separable pattern from O(n6) to O(n5 log n).

Nonhamiltonian 3-Connected Cubic Planar Graphs

We establish that every cyclically 4-connected cubic planar graph of order at most 40 is hamiltonian. Furthermore, this bound is determined to be sharp, and we present all nonhamiltonian examples of order 42. In addition we list all nonhamiltonian cyclically 5-connected cubic planar graphs of order at most 52 and all nonhamiltonian 3-connected cubic planar graphs of girth 5 on at most 46 vertices. The fact that all 3-connected cubic planar graphs on at most 176 vertices and with face size at most 6 are hamiltonian is also verified.

A note on the number of graceful labellings of paths

With the help of a simple recursive construction we give a computer-assisted proof that the number of graceful labellings of a path of length n grows asymptotically at least as fast as (5/3)n. Results of this type have found surprising applications in topological graph theory.

Permutations of a Multiset Avoiding Permutations of Length 3

Abstract We consider permutations of a multiset which do not contain certain ordered patterns of length 3. For each possible set of patterns we provide a structural description of the permutations avoiding those patterns, and in many cases a complete enumeration of such permutations according to the underlying multiset.

P 3 -isomorphisms for graphs

The P3-graph of a finite simple graph G is the graph whose vertices are the 3-vertex paths of G, with adjacency between two such paths whenever their union is a 4-vertex path or a 3-cycle. In this paper we show that connected fnite simple graphs G and H with isomorphic P3-graphs are either isomorphic or part of three exceptional families. We also characterize all isomorphisms between P3-graphs in terms of the original graphs.

M -alternating paths in n -extendable bipartite graphs

Let G be a bipartite graph with bipartition (X,Y) which has a perfect matching. It is proved that G is n-extendable if and only if for any perfect matching M of G and for each pair of vertices x in X and y in Y there are n internally disjoint M-alternating paths connecting x and y. Furthermore, these n paths start and end with edges in E(G)\M. This theorem is then generalized.

On the matching extendability of graphs in surfaces

A graph G with at least 2n+2 vertices is said to be n-extendable if every matching of size n in G extends to a perfect matching. It is shown that (1) if a graph is embedded on a surface of Euler characteristic @g, and the number of vertices in G is large enough, the graph is not 4-extendable; (2) given g>0, there are infinitely many graphs of orientable genus g which are 3-extendable, and given g@?>=2, there are infinitely many graphs of non-orientable genus g@? which are 3-extendable; and (3) if G is a 5-connected triangulation with an even number of vertices which has genus g>0 and sufficiently large representativity, then it is 2-extendable.

Cycles Through 23 Vertices in 3-Connected Cubic Planar Graphs

Abstract. We establish that if A is a set of at most 23 vertices in a 3-connected cubic planar graph G, then there is a cycle in G containing A. This result is sharp.

On matching extensions with prescribed and proscribed edge sets II

Abstract Let G be a graph with at least 2( m + n + 1) vertices. Then G is E ( m , n ) if for each pair of disjoint matchings M , N ⊆ E ( G ) of size m and n , respectively, there exists a perfect matching F in G such that M ⊆ F and F ∩ N = O ;. In this paper, we extend previous results due to Chen (Discrete Math., to appear) as well as results of the present authors (Aldred et al., Discrete Math., to appear) concerning the property E ( m , n ). The first extends a result on claw-free graphs and the second generalizes a result about bipartite graphs.