有许多洞但竞争数小的图
Graphs having many holes but with small competition numbers
摘要：
图G的竞争数k（G）是使G加上k个孤立点成为无圈有向图的竞争图的最小数k。图的长度至少为4的无弦圈称为图的洞。一个图的洞数与它的竞争数密切相关，因为不含洞的图的竞争数至多为1，而有洞的完全二部图$K_{\lfloor \frac{n}{2} \rfloor, \lceil \frac{n}{2} \rceil}$的竞争数是n阶图中最大的.本文证明了即使一个连通图G有许多洞，在一定的假设下，G的竞争数也可以小到2。另外，我们证明了对于一个连通图G，如果G的洞数是两两边不相交的，且G的团数$ω= ω(G)$满足$2 \leq ω\leq h+1$，则G的竞争数至多为$h - ω+ 3$ .
英文摘要：
The competition number k(G) of a graph G is the smallest number k such that G together with k isolated vertices added is the competition graph of an acyclic digraph. A chordless cycle of length at least 4 of a graph is called a hole of the graph. The number of holes of a graph is closely related to its competition number as the competition number of a graph which does not contain a hole is at most one and the competition number of a complete bipartite graph $K_{\lfloor \frac{n}{2} \rfloor, \lceil \frac{n}{2} \rceil}$ which has so many holes that no more holes can be added is the largest among those of graphs with n vertices. In this paper, we show that even if a connected graph G has many holes, the competition number of G can be as small as 2 under some assumption. In addition, we show that, for a connected graph G with exactly h holes and at most one non-edge maximal clique, if all the holes of G are pairwise edge-disjoint and the clique number $ω= ω(G)$ of G satisfies $2 \leq ω\leq h+1$, then the competition number of G is at most $h - ω+ 3$.
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。
