Journal of Graph Theory最新文献

A super stable tensegrity introduced by Connelly in 1982 is a globally rigid discrete structure made from stiff bars and struts connected by cables with tension. We introduce the super stability number of a multigraph as the maximum dimension that a multigraph can be realized as a super stable tensegrity, and show that it equals the Colin de Verdière number minus one. As a corollary we obtain a combinatorial characterization of multigraphs that can be realized as three-dimensional super stable tensegrities. We also show that, for any fixed , there is an infinite family of 3-regular graphs that can be realized as -dimensional injective super stable tensegrities.

We study a counting version of Cycle Double Cover Conjecture. We discuss why it is more interesting to count circuits (i.e., graphs isomorphic to $��\; ��C_k��$ for some $��\; ��k��$) instead of cycles (graphs with all degrees even). We give an almost-exponential lower bound for graphs with a surface embedding of representativity at least 4. We also prove an exponential lower bound for planar graphs. We conjecture that any bridgeless cubic graph has at least $��\; ��2^{�n�\; �/�\; �2�\; �-�\; �1�}��$ circuit double covers and we show an infinite class of graphs for which this bound is tight.

In this note it is proven that there exist infinitely many positive integers $��a�$ and $��b�$ such that the Cartesian product of a directed cycle of length $��2�\; �a�$ and a directed cycle of length $��2�\; �b�$ is non-Hamiltonian. In particular, the Cartesian product of an 880-dicycle and a 4368-dicycle is non-Hamiltonian. We also prove that there is no such graph on fewer than $��880�\; �\cdot �\; �4368�\; �=�\; �3�\; �,�\; �843�\; �,�\; �840�$ vertices, which is rather astonishing.

In recent years, discrete notions of curvature have been defined and exploited to understand various geometric properties of graphs; especially regarding heat flow, and spectral properties. In this paper, we study various combinatorial properties implied by satisfying the Bakry–Émery curvature dimension inequality $��\; ��C�\; �D�\; ��\left(�\; ��\infty �\; �,�\; �K��\; �\right)���$. In particular we derive a local discrepancy inequality, similar in spirit to the expander mixing lemma from spectral graph theory, which certifies a type of “local pseudo-randomness” of the edge set of the graph, for graphs satisfying a curvature lower bound. In addition, several other consequences are derived regarding graph connectivity and cycle statistics of the graph.