A $k$-majority tournament on a finite vertex set is defined by $2k-1$ linear orders, with $u\to v$ when $u$ lies above $v$ in at least $k$ of the orders. Let $F(k)$ be the maximum, over all $k$-majority tournaments, of the size of a minimum dominating set. Alon, Brightwell, Kierstead, Kostochka, and Winkler proved that $C_1k/\log k \leq F(k) \leq C_2k\log k$ for suitable positive constants $C_1$ and $C_2$. In this paper, we prove the linear lower bound $F(k)\ge \left\lfloor\frac{k+1}{2}\right\rfloor $ for $k\ge 3$.
For integers $n,r,t$ with $2\le t\le r-1$, let $f(n,r,t+1)$ denote the largest possible minimum degree of a balanced $r$-partite graph with parts of size $n$ and containing no copy of $K_{t+1}$. Lo, Treglown and Zhao identified $f(n,7,4)$ as the only remaining case in their treatment of the $K_4$-free family. I determi...
For a positive integer $k$, let $\beta_k(G)$ be the maximum cardinality of a vertex set inducing maximum degree less than $k$, and let $SF_k(G)$ be the maximum cardinality of a minimal $k$-star-forming set. The known inequality $\beta_k(G)\le SF_k(G)$ suggests asking when equality holds. We place this question in the f...
A dominating set $D$ of a graph $G$ is called $k$-limited if every vertex of $D$ has at most $k$ neighbors outside $D$. The minimum cardinality among all $k$-limited dominating sets of $G$ is the $k$-limited domination number, denoted by $\gamma_k^L(G)$. In this paper, we prove that the $1$-Limited Dominating Set probl...
For a family $\mathcal{F}$ of $k$-graphs, $\ex_k(n,\mathcal{F})$ denotes the maximum number of edges in an $n$-vertex $\mathcal{F}$-free $k$-graph. Let $M_{s+1}^k$ denote a matching of size $s+1$ in $k$-uniform hypergraphs. Recently, Alon and Frankl (JCTB, 2024) determined $\ex_2(n,\{M_{s+1}^2,K_{\ell+1}\})$ for all $n...
For a tournament $T$, let $\omega(T)$ be the minimum clique number among the backedge graphs of $T$, and let $\chi(T)$ be its dichromatic number. We give a template-lifting construction. It turns a $k$-template into a regular, vertex-transitive Cayley tournament that is simultaneously $(k+1)$-$\omega$-critical and $(k+...
For a graph $F$ with at least one edge, a $k$-uniform hypergraph is Berge-$F$-saturated if it contains no Berge-$F$, but adding any missing hyperedge creates a Berge-$F$. The saturation number $\text{sat}_k(n,\text{Berge-}F)$ is the minimum number of edges in such a hypergraph on $n$ vertices. English, Gordon, Graber,...
Tian-Ying Xie· 0 citations
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