Journal of Graph Theory最新文献

An independent transversal (IT) in a graph $��G�$ with a given vertex partition $��P�$ is an independent set of vertices of $��G�$ (i.e., it induces no edges), that consists of one vertex from each part (block) of $��P�$. Over the years, various criteria have been established that guarantee the existence of an IT, often given in terms of $��P�$ being $��t�$-thick, meaning all blocks have size at least $��t�$. One such result, obtained recently by Wanless and Wood, is based on the maximum average block degree $��b��\left(��G�,�P��\right)��=�\max ��\{��{\sum }_{�u�\in �U�}�d��$

Erdős and Wilson proved in 1977 that almost all graphs have chromatic index equal to their maximum degree. In 2001 Balister extended this result and proved that the same number of colours is almost always sufficient if we additionally demand the distinctness of the sets of colours incident with any two vertices. We study a stronger condition and show that one more colour is almost always sufficient and necessary if the inclusion of these sets is forbidden for any pair of adjacent vertices. We also settle the value of a more restrictive graph invariant for almost all graphs, where inclusion is forbidden for all pairs of vertices, which necessitates one more colour for graphs of even order.

In 2019, Dvořák asked whether every connected graph $��G�$ has a tree decomposition $��\left(�\; ��T�\; �,�\; �B��\; �\right)�$ so that $��T�$ is a subgraph of $��G�$ and the width of $��\left(�\; ��T�\; �,�\; �B��\; �\right)�$ is bounded by a function of the treewidth of $��G�$. We prove that this is false, even when $��G�$ has treewidth 2 and $��T�$ is allowed to be a minor of $��G�$.

We consider the minimum degree stability of graphs forbidding odd cycles: What is the tight bound on the minimum degree to guarantee that the structure of a $��C_{�2�\; �k�\; �+�\; �1�}�$-free graph inherits from the extremal graph (a balanced complete bipartite graph)? Andrásfai, Erdős, and Sós showed that if a $��\{�\; �C_3�\; �,�\; �C_5�\; �,�\; �\text{…}�\; �,�\; �C_{�2�\; �k�\; �+�\; �1�}�\; �\}�$-free graph on $��n�$ vertices has minimum degree greater than $��\frac{2}{�2�\; �k�\; �+�\; �3�}�\; �n�$, then it is bipartite. Häggkvist showed that for $��$

A triangle-path in a graph $��G�$ is a sequence of distinct triangles $��T_1�\; �,�\; �T_2�\; �,�\; �\dots �\; �,�\; �T_m�$ in $��G�$ such that for any $��i�\; �,�\; �j�$ with $��1�\; �\le �\; �i�\; �<�\; �j�\; �\le �\; �m�$,