Journal of Graph Theory最新文献

It is known that the flip distance between two triangulations of a convex polygon is related to the smallest number of tetrahedra in the triangulation of some polyhedron. The latter was used to compute the diameter of the flip graph of convex polygons with a large number of vertices. However, it is yet unknown whether the flip distance and this smallest number of tetrahedra are always the same or even close. In this work, we find examples to show that the ratio between these two numbers can be arbitrarily close to $��\frac{3}{2}�$. We also propose two conjectures in the end, one about this ratio, and the other may have some implications on when two triangulations can achieve maximal distance.

A 2-factor in a graph $��G�$ is a subset of edges $��M�$ such that every node of $��G�$ is incident with exactly two edges of $��M�$. Many results are known concerning 2-factors including a polynomial-time algorithm for finding 2-factors and a characterization of those graphs that have a 2-factor. The problem of finding a 2-factor in a graph is a relaxation of the NP-hard problem of finding a Hamilton cycle. A stronger relaxation is the problem of finding a triangle-free 2-factor, that is, a 2-factor whose edges induce no cycle of length 3. In this paper, we present a polynomial-time algorithm for the problem of finding a triangle-free 2-factor as well as a characterization of the graphs that have such a 2-factor and related min–max and augmenting path theorems.

In this paper, we study the Wiener index of the orientation of trees and theta-graphs. An orientation of a tree is called no-zig-zag if there is no subpath in which edges change the orientation twice. Knor, Škrekovski, and Tepeh conjectured that every orientation of a tree $��T�$ achieving the maximum Wiener index is no-zig-zag. We disprove this conjecture by constructing a counterexample. Knor, Škrekovski, and Tepeh conjectured that among all orientations of the theta-graph $��{\Theta }_{�a�\; �,�\; �b�\; �,�\; �c�}�$ with $��a�\; �\ge �\; �b�\; �\ge �\; �c�$ and $��b�\; �>�\; �1�$, the maximum Wiener index is achieved by the one in which the union of the paths between $��u_1�$ and $��u_2�$ forms a directed cycle of length $��a�\; �+�\; �b�\; �+�\; �$

We study the minimum number of maximum matchings in a bipartite multigraph $��G�$ with parts $��X�$ and $��Y�$ under various conditions, refining the well-known lower bound due to M. Hall. When $��\mid �X�\mid �=�n�$, every vertex in $��X�$ has degree at least $��k�$, and every vertex in $��X�$ has at least $��r�$ distinct neighbors, the minimum is $��r�!��\left(��k�-�r�+�1��\right)��$ when $��n�\ge �r�$ and is $���[��r�$

Given a planar graph family $��F�$, let $��e�x_P��(�n�,�F�)��$ and $��s�p�e�x_P��(�n�,�F�)��$ be the maximum size and maximum spectral radius over all $��n�$-vertex $��F�$-free planar graphs, respectively. Let $��t�C_{\ell }�$ be the disjoint union of $��t�$ copies of $��\ell �$-cycles, and $��t�C�$ be the family of $�$

We study the weak $��K_s�$-saturation number of the Erdős–Rényi random graph $��G��\left(��n�\; �,�\; �p��\; �\right)��$, denoted by $��\text{wsat}��\left(��G��\left(��n�\; �,�\; �p��\; �\right)��\; �,�\; �K_s��\; �\right)��$, where $��K_s�$ is the complete graph on $��s�$ vertices. In 2017, Korándi and Sudakov proved that the weak $��K_s�$-sat

A digraph $��G�$ is $��k�$-geodetic if for any (not necessarily distinct) vertices $��u�,�v�$ there is at most one directed walk from $��u�$ to $��v�$ with length not exceeding $��k�$. The order of a $��k�$-geodetic digraph with minimum out-degree $��d�$ is bounded below by the directed Moore bound $��M��\left(��d�,�k��\right)��=�1�+�d�+�d^2�+�\cdots �+�d^k�$

We prove that for the $��d�$-regular tessellations of the hyperbolic plane by $��k�$-gons, there are exponentially more self-avoiding walks of length $��n�$ than there are self-avoiding polygons of length $��n�$. We then prove that this property implies that the self-avoiding walk is ballistic, even on an arbitrary vertex-transitive graph. Moreover, for every fixed $��k�$, we show that the connective constant for self-avoiding walks satisfies the asymptotic expansion $��d�\; �-�\; �1�\; �-�\; �O�\; ��\left(�\; ��1�\; �∕�\; �d��\; �\right)��$ as $��d�\; �\to �\; �\infty �$; on the other hand, the connective constant for self-avoiding polygons remains bounded. Finally, we show for all but two tessellations that the number of self-avoiding walks of length

The prism over a graph $��G�$ is the Cartesian product of $��G�$ with the complete graph on two vertices. A graph $��G�$ is prism-hamiltonian if the prism over $��G�$ is hamiltonian. We prove that every polyhedral graph (i.e., 3-connected planar graph) of minimum degree at least four is prism-hamiltonian.

Let $��G�$ be a simple graph with maximum degree $��\Delta ��\left(�G�\right)��$. A subgraph $��H�$ of $��G�$ is overfull if $��\mid �E��\left(�H�\right)��\mid �>�\Delta ��\left(�G�\right)���\lfloor ��\frac{1}{2}�\mid �V��\left(�H�\right)��\mid ��\rfloor ��$. Chetwynd and Hilton in 1986 conjectured that a graph $��G�$ with