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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 = ∅. In the present paper we wish to study property E(m, n) for the various values of integers m and n when the graphs in question are(More)
Two parties A and B select a cards and b cards from a known deck and a third party C receives the remaining c cards. We consider methods whereby A can, in a single message, publicly inform B of her hand without C learning any card held by A or by B. Conditions on a, b, c are given for the existence of an appropriate message.
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(More)
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 such graphs 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(More)
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. Let G be a 3-connected cubic planar graph and let A ⊆ V (G). It was shown in [4] that if |A| ≤ 19 there is a cycle C in G such that A ⊆ V (C). In this paper we show that if |A| ≤ 23, then G contains a(More)