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\iteman{io-port 06104595}
\itemau{Wang, Jun; Zhang, Huajun}
\itemti{Nontrivial independent sets of bipartite graphs and cross-intersecting families.}
\itemso{J. Comb. Theory, Ser. A 120, No. 1, 129-141 (2013).}
\itemab
Summary: Let $G(X,Y)$ be a connected, non-complete bipartite graph with $|X|\leqslant |Y|$. An independent set $A$ of $G(X,Y)$ is said to be trivial if $A\subseteq X$ or $A\subseteq Y$. Otherwise, $A$ is nontrivial. By $\alpha (X,Y)$ we denote the maximum size of nontrivial independent sets of $G(X,Y)$. We prove that if the automorphism group of $G(X,Y)$ is transitive and primitive on $X$ and $Y$, respectively, then $\alpha (X,Y)=|Y| - d(X)+1$, where $d(X)$ is the degree of vertices in $X$. We also give the structures of maximum-sized nontrivial independent sets of $G(X,Y)$. Consequently, these results give the sizes and structures of maximum-sized cross-$t$-intersecting families of finite sets, finite vector spaces and permutations, as well as the sizes and structures of maximum-sized cross-Sperner families of finite sets and finite vector spaces.
\itemrv{~}
\itemcc{}
\itemut{intersecting family; cross-intersecting family; symmetric system; Erd\H{o}s-Ko-Rado theorem}
\itemli{doi:10.1016/j.jcta.2012.07.005}
\end