Yoshiyasu Ishigami

Vertex-Disjoint Cycles Containing Specified Edges*

Abstract. Dirac and Ore-type degree conditions are given for a graph to contain vertex disjoint cycles each of which contains a previously specified edge. One set of conditions is given that imply vertex disjoint cycles of length at most 4, and another set of conditions are given that imply the existence of cycles that span all of the vertices of the graph (i.e. a 2-factor). The conditions are shown to be sharp and give positive answers to conjectures of Enomoto in [3] and Wang in [5].

The wide-diameter of the n-dimensional toroidal mesh

In graph theory and a study of transmission delay and fault tolerance of networks, the connectivity and the diameter of a graph are very important and they have been studied by many mathematicians. We studied the wide-diameter of a k-regular k-connected graph which is defined by the maximum of the k-distance between two distinct vertices, when the k-distance between x and y is equal to the least number l such that there exists k vertex-disjoint paths between x and y whose lengths are at most l. Because the wide-diameter of any k-regular k-connected graph is greater than its diameter, a k-regular k-connected graph whose wide-diameter is equal to “1 + its diameter” is optimal. For example, it is known that the hypercube is such a graph. We define n-dimensional toroidal mesh C(d1, d2, ………, dn) with vertices {(x1, ………, xn)|0 ≤ xi < di (1 ≤ i ≤ n)}. Each vertex (x1, ………, xn) is adjacent to 2n other vertices: (x1 ± 1, x2, ………, xn), (x1, x2 ± 1, ………, xn), ………, (x1, x2, ………, xn ± 1), where additions are performed modulo di (1 ≤ i ≤ n). This graph is an n-dimensional orthogonal mesh with global edges, which is 2n-connected and 2n-regular. We show that the graph satisfies the above property with respect to its 2n-diameter and its diameter except for some special cases.

Linear Ramsey Numbers for Bounded-Degree Hypergrahps

Abstract We show that the Ramsey number is linear for every uniform hypergraph with bounded degree. This is a hypergraph extension of the famous theorem for ordinary graphs which Chvatal et al. [V. Chvatal, V. Rodl, E. Szemeredi and W.T. Trotter, Jr., The Ramsey number of a graph with bounded maximum degree, J. Combin. Theory Ser. B 34 (1983), pp. 239–243] showed in 1983. Our proof demonstrates the potential of a new regularity lemma by [Y. Ishigami, A simple regularization of hypergraphs, preprint, arXiv:math/0612838 (2006)].

VC-dimensions of finite automata and commutative finite automata with k letters and n states

Abstract An investigation is conducted of the Vapnik-Chervonenkis dimensions (VC-dimensions) of finite automata having k letters and n states. It is shown for a fixed positive integer k (⩾ 2), that (1) the VC-dimension of DFA k (n):= {L ⊂ {1, 2, …, k}∗ : some deterministic finite automaton with at most n states accepts L} is n + log2 n − O(log logn) for k = 1 and (k − 1 + 0(1))n log2 n for k ⩾ 2, and (2) the VC-dimension of CDFAk(n):= {L ϵ DFAk(n) : L is commutative} is n + o(n).

Proof of a Conjecture of Bollobás and Kohayakawa on the Erdoős-Stone Theorem

For any integer r?1, let a(r) be the largest constant a?0 such that if ?>0 and 00 for all r and conjectured that liminfr?∞a(r)?0. V. Chvatal and E. Szemeredi (1981, J. London Math. Soc. (2)23, 207?214) settled it by giving a(r)?0.002 for all r. We show that, for all r,a(r)=a(1). Further we prove the conjecture of B. Bollobas and Y. Kohayakawa (1994, Combinatorica14, 279?286). The weak form of it states that for any r?1, 00. That is, all color classes but one are relatively large for fixed r, small c?0, and large n?∞. Our proof method is based on Szemeredis Regularity Lemma.

An extension of a theorem on cycles containing specified independent edges

We give an alternative proof of a conjecture due to Wang (J. Graph Theory 26 (1997) 105) in a stronger form. The main theorem states that for any integer k ≥ 2 if G is a graph of order n ≥ 4k - 1 and d(u) + d(υ) ≥ n + 2k - 2 for each pair of non-adjacent vertices u and υ of G, then, for any k independent edges e1,... ,ek of G, there exist k vertex-disjoint cycles C1,..., Ck in G such that (i) ei ∈ E(Ci) for all 1 ≤ i ≤ k, (ii) V(C1)∪ ... ∪V(Ck) = V(G), and (iii) #{i ≤ k||Ci| a > n - 4k + 2). It strengthens the conjecture of Wang, which was first proven by Egawa et al. (Graphs Combin. 16 (2000) 81).

Containment problems in high-dimensional spaces

AbstractFor any integersn, d ≥ 2, letП(n, d) be the largest number such that every setP ofn points inRd contains two pointsx, y ∈ P satisfying |boxd(x, y) ∩ P| ≥П(n, d), where boxd(x, y) means the smallest closed box with sides parallel to the axes, containingx andy. We show that, for any integersn,

An extremal problem of d permutations containing every permutation of every t elements

d \geqslant 2,\frac{2}{{(2\sqrt 2 )^{2^d } }}n + 2 \leqslant \prod (n,d) \leqslant \frac{2}{{7^{[d/5]} 2^{2^{d - 1} } }}n + 5