Journal of Graph Theory最新文献

This paper proves that every planar graph having no cycle of length 4, 7, or 9 is DP-3-colorable.

Let $��G�\; �≔��(�\; �G_1�\; �,�\; �G_2�\; �,�\; �G_3�\; �)��$ be a triple of graphs on a common vertex set $��V�$ of size $��n�$. A rainbow triangle in $��G�$ is a triple of edges $���\left(��e_1�\; �,�\; �e_2�\; �,�\; �e_3��\; �\right)��$

Let $��N_P�\; ��\left(�\; ��n�\; �,�\; �H��\; �\right)��$ denote the maximum number of copies of $��H�$ in an $��n�$ vertex planar graph. The problem of bounding this function for various graphs $��H�$ has been extensively studied since the 70's. A special case that received a lot of attention recently is when $��H�$ is the path on $��2�\; �m�\; �+�\; �1�$ vertices, denoted $��P_{�2�\; �m�\; �+�\; �1�}�$. Our main result in this paper is that

This improves upon the previously best known

We introduce a supporting combinatorial framework for the Flat Wall Theorem. In particular, we suggest two variants of the theorem and we introduce a new, more versatile, concept of wall homogeneity as well as the notion of regularity in flat walls. All proposed concepts and results aim at facilitating the use of the irrelevant vertex technique in future algorithmic applications.

A digraph $��H�$ is called ubiquitous if every digraph $��D�$ that contains $��k�$ vertex-disjoint copies of $��H�$ for every $��k�\in �N�$ also contains infinitely many vertex-disjoint copies of $��H�$. We characterise which digraphs with rays as underlying undirected graphs are ubiquitous.

We consider Erdős-Rényi graphs $��G��\left(��n�,�p��\right)��$ for $��0�<�p�<�1�$ fixed and $��n�\to �\infty �$ and study the expected number of steps, $��H_{�w�v�}�$, that a random walk started in $��w�$ needs to first arrive in $��v�$. A natural guess is that an Erdős-Rényi random graph is so homogeneous that it does not really distinguish between vertices and $��H_{�w�v�}�=��\left(��1�+�o��\left(�1�\right)���\right)��n�$