二分图中支配集数量的诺德豪斯 - 加达姆不等式
Nordhaus-Gaddum Inequalities for Dominating-Set Counts in Bipartite Graphs
AI总结：
研究二分图中支配集数量的诺德豪斯 - 加达姆不等式，通过证明更强的界\(\partial(G) + \partial(\bar{G}) \leq 2(2^{|A|} - 1)(2^{|B|} - 1) + 2\)部分解决猜想，并刻画了等式成立的二分图。
AI中文摘要：
图\(G\)中的支配集\(S\)是其顶点的一个子集，使得\(G\)中的每个顶点要么在\(S\)中，要么与\(S\)中的一个顶点相邻。诺德豪斯 - 加达姆不等式关联了图及其补图上的一个图参数的值。在此情形下，基奥和谢恩猜想任何\(n\)个顶点的图\(G\)满足\(\partial(G) + \partial(\bar{G}) \leq 2(2^{\lfloor n/2 \rfloor} - 1)(2^{\lceil n/2 \rceil} - 1) + 2\)，其中\(\partial(G)\)是\(G\)中支配集的数量。我们通过证明更强的界：对于具有非空二分划\((A,B)\)的二分图\(G\)，有\(\partial(G) + \partial(\bar{G}) \leq 2(2^{|A|} - 1)(2^{|B|} - 1) + 2\)，部分解决了该猜想。我们还刻画了等式成立的二分图。
英文摘要：
A dominating set in a graph $G$ is a subset $S$ of its vertices such that each vertex in $G$ is either in $S$ or adjacent to a vertex in $S$. Nordhaus-Gaddum inequalities relate the values of a graph parameter on a graph and its complement. In this setting, Keough and Shane conjecture that any graph $G$ on $n$ vertices satisfies $\partial(G) + \partial(\bar{G}) \leq 2(2^{\lfloor n/2 \rfloor} - 1)(2^{\lceil n/2 \rceil} - 1) + 2$, where $\partial(G)$ is the number of dominating sets in $G$. We partially resolve this conjecture for the bipartite case by proving the stronger bound: for a bipartite graph $G$ with nonempty bipartition $(A,B)$, it holds that $\partial(G) + \partial(\bar{G}) \leq 2(2^{|A|} - 1)(2^{|B|} - 1) + 2$. We also characterize the bipartite graphs for which equality holds.
