A partition of the edges of a graph G into sets {S 1 , . . . , S k } defines a multiset X v for each vertex v where the multiplicity of i in X v is the number of edges incident to v in S i . 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, X u = X v . 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.

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.

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 provides such improvements, though these are 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(n 6 ) to O(n 5 log n).

Abstract:In this paper, we consider forbidden subgraphs which force the existence of a 2-factor. Let G be the class of connected graphs of minimum degree at least two and maximum degree at least three, and let F 2 be the class of graphs which have a 2-factor. For a set H of connected graphs of order at least three, a graph G is said to be H-free if no member of H is an induced subgraph of G, and let G(H) denote the class of graphs in G that are H-free. We are interested in sets H such that G(H) is an infinite class while G(H)−F 2 is a finite class. In particular, we investigate 250 FORBIDDEN SUBGRAPHS AND THE EXISTENCE OF A 2-FACTOR 251whether H must contain a star (i.e. K 1,n for some positive integer n). We prove the following. For |H| ≤ 2, we compare our results with a recent result by Faudree et al. (Discrete Math 308 (2008), 1571-1582, and report a difference in the conclusion when connected graphs are considered as opposed to 2-connected graphs. We also observe a phenomenon in which H does not contain a star but G(H)−G({K 1,t }) is finite for some t ≥ 3.

Abstract. 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.Key words. nonhamiltonian, cubic, planar AMS subject classification. 05C38 PII. S0895480198348665 1. Introduction. In this paper we describe an investigation (making much use of computation) of cyclically k-connected cubic planar graphs (CkCP s) for k = 4, 5 and report the results. We shall also have occasion to consider cubic 3-connected planar graphs with no restriction on cyclic connectivity; these we refer to as C3CP s. The investigation extends the work of Holton and McKay in [10], in which the smallest order of a nonhamiltonian C3CP was shown to be 38. We provide answers to two questions raised in that paper, namely, the following:(a) What is the smallest order of a nonhamiltonian C4CP? (b) Is there more than one nonhamiltonian C5CP on 44 vertices? The answer to the former question is determined to be 42, and all nonhamiltonian C4CPs of that order are presented. The latter question is answered in the negative, and a complete list of all nonhamiltonian C5CPs on at most 52 vertices is presented.Before proceeding, we include some definitions and results from the existing literature as we shall make use of them in the rest of the paper. By a k-gon we mean a face of a plane graph bounded by k edges. Note that a k-cycle is not necessarily a k-gon. By a k-cut we mean a set of k edges whose removal leaves the graph disconnected and of which no proper subset has that property. The two components (and clearly there are only two) formed by the removal of a k-cut are called k-pieces. A k-cut is nontrivial if each of its k-pieces contains a cycle and is essential if it is nontrivial and each of its k-pieces contains more than k vertices. A cubic graph is cyclically k-connected if it has no nontrivial t-cuts for 0 ≤ t ≤ k − 1, and it has cyclic connectivity k if, in addition, it has at least one nontrivial k-cut. We denote by λ (G) the value of k such that the C3CP G has cyclic connectivity k.If G is a hamiltonian C3CP, then an a-edge is an edge which is present in every hamiltonian cycle in G, while a b-edge is absent from every hamiltonian cycle in G.To focus our search for nonhamiltonian C4CPs of smallest order we shall make use of the following theorem which formed the main result in [10].