A graph is said to be reconstructible if it is determined up to isomorphism from the collection of all its one-vertex deleted unlabeled subgraphs. Reconstruction Conjecture (RC) asserts that all graphs on at least three vertices are reconstructible. In this paper, we prove that interval-regular graphs and some new classes of graphs are reconstructible and show that RC is true if and only if all...

In this paper, we show that for any even integer t ≥ 4, every 3-connected graph with no K3,t-minor has a spanning tree whose maximum degree is at most t − 1. This result is a common generalization of the result by Barnette [1] and the one by Chen, Egawa, Kawarabayashi, Mohar and Ota [4].

We show that every core graph with a primitive automorphism group has the property that whenever it is a retract of a product of connected graphs, it is a retract of a factor. The example of Kneser graphs shows that the hypothesis that the factors are connected is essential. In the case of complete graphs, our result has already been shown in [4, 17], and it is an instance where Hedetniemi’s co...

The Calogero equation is used to illustrate the following two aspects of the Painlevé analysis of PDEs: (i) the singular expansions of solutions around characteristic hypersurfaces are neither single-valued functions of independent variables nor single-valued functionals of data; (ii) the truncated singular expansions not necessarily lead to the simplest, elementary, Bäcklund autotransformation...

AbstraeL The nonrevisiting path conjecture for polytopes, which is equivalent to the Hirsch conjecture, is open. However, for surfaces, the nonrevisiting path conjecture is known to be true for polyhedral maps on the sphere, projective plane, toms, and a Klein bottle. Barnette has provided counterexamples on the orientable surface of genus 8 and nonorientable surface of genus 16. In this note t...

We demonstrate that in a certain gauge the elliptic Ruijsenaars– Schneider models admit Lax representation governed by the same dynamical r– matrix as their non–relativistic counterparts (Calogero–Moser models). This phenomenon was previously observed for the rational and hyperbolic models.

We prove that an excluded minor for the class of GF(q)-representable matroids cannot contain a large projective geometry over GF(q) as a minor. © 2005 Elsevier Inc. All rights reserved.

In this article, we give new proofs on the some cases on Weinstein conjecture and get some new results on Weinstein conjecture.