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Let T be a hamiltonian tournament with n vertices and γ a hamiltonian cycle of T. In previous works we introduced and studied the concept of cycle-pancyclism to capture the following question: What is the maximum intersection with γ of a cycle of length k? More precisely, for a cycle Cₖ of length k in T we denote , the number of arcs that γ and Cₖ have in common. Let and f(n,k) = minf(k,T,γ)|T is a hamiltonian tournament with n vertices, and γ a hamiltonian cycle of T. In previous papers we gave...
Almost all d-regular graphs are Hamiltonian, for d ≥ 3 [8]. In this note we conjecture that in a similar, yet somewhat different, sense almost all cubic non-Hamiltonian graphs are bridge graphs, and present supporting empirical results for this prevalence of the latter among all connected cubic non-Hamiltonian graphs.
For any we construct graphs of degree , diameter , and order , obtained as lifts of dipoles with voltages in cyclic groups. For Cayley Abelian graphs of diameter two a slightly better result of has been known [3] but it applies only to special values of degrees depending on prime powers.
Let Ks,t be the complete bipartite graph with partite sets of size s and t. Let L1 = ([a1, b1], . . . , [am, bm]) and L2 = ([c1, d1], . . . , [cn, dn]) be two sequences of intervals consisting of nonnegative integers with a1 ≥ a2 ≥ . . . ≥ am and c1 ≥ c2 ≥ . . . ≥ cn. We say that L = (L1; L2) is potentially Ks,t (resp. As,t)-bigraphic if there is a simple bipartite graph G with partite sets X = {x1, . . . , xm} and Y = {y1, . . . , yn} such that ai ≤ dG(xi) ≤ bi for 1 ≤ i ≤ m, ci ≤ dG(yi) ≤ di for...
A family is constructed of cardinality equal to the continuum, whose members are totally incomparable hereditarily indecomposable Banach spaces.
A special case of a combinatorial theorem of De Bruijn and Erdős asserts that every noncollinear set of points in the plane determines at least distinct lines. Chen and Chvátal suggested a possible generalization of this assertion in metric spaces with appropriately defined lines. We prove this generalization in all metric spaces where each nonzero distance equals or .
The paper provides a proof of a combinatorial result which pertains to the characterization of the set of equations which are solvable in the composition monoid of all partial functions on an infinite set.
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