恰有两个洞的图的竞争数
The competition number of a graph with exactly two holes
摘要：
设D是无圈有向图。D的竞争图是具有与D相同的顶点集且在x和y之间有一条边的图当且仅当存在D中的顶点v使得（x，v）和（y，v）是D的弧。对任意图G，G加上足够多的孤立点是某个无圈有向图的竞争图。图G的竞争数k（G）是图G中孤立点的最小个数。图的洞是作为诱导子图的长度至少为4的圈。2005年，Kim [5]证明了具有h个洞的图的竞争数至多为h+1。虽然李和张[8]和金等人。[7]表明，她的猜想是正确的，当孔不重叠太多，它仍然是开放的情况下，孔共享边缘在一个任意的方式。为了共享一条边，一个图必须至少有两个洞，所以很自然地从一个恰好有两个洞的图开始。本文证明了这一猜想对这类图是成立的。
英文摘要：
Let D be an acyclic digraph. The competition graph of D is a graph which has the same vertex set as D and has an edge between x and y if and only if there exists a vertex v in D such that (x,v) and (y,v) are arcs of D. For any graph G, G together with sufficiently many isolated vertices is the competition graph of some acyclic digraph. The competition number k(G) of G is the smallest number of such isolated vertices. A hole of a graph is a cycle of length at least 4 as an induced subgraph. In 2005, Kim [5] conjectured that the competition number of a graph with h holes is at most h+1. Though Li and Chang [8] and Kim et al. [7] showed that her conjecture is true when the holes do not overlap much, it still remains open for the case where the holes share edges in an arbitrary way. In order to share an edge, a graph must have at least two holes and so it is natural to start with a graph with exactly two holes. In this paper, the conjecture is proved true for such a graph.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
