Journal of Graph Theory最新文献

For a $��k�$-uniform hypergraph $��F�$ we consider the parameter $��\Theta ��\left(�\; �F�\; �\right)��$, the minimum size of a clique cover of the edge set of $��F�$. We derive bounds on $��\Theta ��\left(�\; �F�\; �\right)��$ for $��F�$ belonging to various classes of hypergraphs.

Gyárfás and Sumner independently conjectured that for every tree $��T�$, there exists a function $��f_T�\; �:�\; �N�\; �\to �\; �N�$ such that every $��T�$-free graph $��G�$ satisfies $��\chi ��\left(�\; �G�\; �\right)��\; �\le �\; �f_T��\left(��\omega ��\left(�\; �G�\; �\right)���\; �\right)��$, where $��\chi ��\left(�\; �G�\; �\right)��$ and $��\omega ��\left(�\; �G�\; �\right)�$

A Kempe chain on colors $��a�$ and $��b�$ is a component of the subgraph induced by colors $��a�$ and $��b�$. A Kempe change is the operation of interchanging the colors of some Kempe chains. For a list-assignment $��L�$ and an $��L�$-coloring $��\phi �$, a Kempe change is $��L�$-valid for $��\phi �$ if performing the Kempe change yields another $��L�$-coloring. Two $��L�$-colorings are $��L�$-equivalent if we can form one from the other by a sequence of $�$

For a $��k�$-tree $��T�$, a generalization of a tree, the local mean order of sub-$��k�$-trees of $��T�$ is the average order of sub-$��k�$-trees of $��T�$ containing a given $��k�$-clique. The problem whether the maximum local mean order of a tree (i.e., a 1-tree) at a vertex is always taken on at a leaf was asked by Jamison in 1984 and was answered by Wagner and Wang in 2016. Actually, they proved that the maximum local mean order of a tree at a vertex occurs either at a leaf or at a vertex of degree 2. In 2018, Stephens and Oellermann asked a similar problem: for any $��k�$-tree $��T�$, does the maximum local mean order of sub-$��k�$-trees containing a given $��k�$-clique occur at a $��k�$-clique that is not a major