《毛毛虫与龙虾》中的星星
On Stars in Caterpillars and Lobsters
摘要：
包含一个固定顶点$v$的图的所有$k$ -独立集的族称为{星}，$v$称为它的中心。星是有趣的，因为它们与Erdös-Ko-Rado图的关系。Hurlbert和Kamat证明，在树上，最大的星星集中在叶子上。这一猜想被Baber、Borg、Feghali、Johnson和Thomas独立地证明是错误的。在本文中，我们介绍了一个工具，以约束的大小，以恒星为中心的恒星在某些顶点的叶子。我们用这个工具来表明，毛毛虫和太阳图满足赫尔伯特和卡玛特的猜想，并表明，在龙虾中最大的恒星的中心要么叶或脊椎顶点的程度2。
英文摘要：
The family of all $k$-independent sets of a graph containing a fixed vertex $v$ is called a {star} and $v$ is called its center. Stars are interesting for their relation to Erdös-Ko-Rado graphs. Hurlbert and Kamat conjectured that in trees the largest stars are centered in leafs. This conjecture was disproven independently by Baber, Borg, and Feghali, Johnson, and Thomas. In this paper we introduce a tool to bound the size of stars centered at certain vertices by stars centered at leafs. We use this tool to show that caterpillars and sunlet graphs satisfy Hurlbert and Kamat's conjecture, and to show that the centers of the largest stars in lobsters are either leafs or spinal vertices of degree 2.
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