Let \(H\) be a finite hypergraph with rank $r$ and minimum degree \(d\). In the Maker--Breaker degree game, Maker and Breaker alternately claim previously unclaimed hyperedges of \(H\), with Maker moving first, and Breaker seeks to maximize the minimum degree $\delta(H_{\mathrm B})$ of his spanning subhypergraph $H_{\mathrm B}$. We prove that, for every fixed \(r\ge2\) and all sufficiently large \(d\), Breaker has a deterministic strategy satisfying \[ \delta(H_{\mathrm B})\ge \frac{d}{r+1}-\sqrt{d\log d}. \] For graphs, this improves the classical universal lower bound \(d/4\) to \(d/3-\sqrt{d\log d}\). The proof introduces a virtual balancing process, encodes local imbalance by a multiplicative risk, and keeps the resulting risk vector in the Shearer region throughout the game.
For $H$ a hypergraph, the Hall ratio $\rho(H)=\text{max}_{X\subseteq V(H)}\frac{\lvert{X}\rvert}{\alpha(H[X])}$ is a natural lower bound for its chromatic number, and among non-empty $k$-uniform hypergraphs the smallest possible Hall ratio is $k/(k-1)$. For graphs ($k = 2$), having minimum Hall ratio $2$ forces chromat...
Veronica Bitonti, Freddie Illingworth, Pedram Noohi et al.· 0 citations
We prove a sparse version of the four-colour theorem of Brandt and Thomass\'{e}, answering a question of Allen, B\"ottcher, Kohayakawa and Roberts. For every fixed $0<\gamma\le1/10$ and every $p=p(n)\in(0,1]$, asymptotically almost surely every spanning triangle-free $H\subseteq G(n,p)$ with $\delta(H)\ge(1/3+\gamma)pn...
The toughness of a connected noncomplete graph $G$ is$$\tau(G)=\min\left\{\frac{|S|}{c(G-S)}:S\subseteq V(G),\ c(G-S)\geq2\right\},$$where $c(G)$ is the number of components of $G$; as usual, $\tau(K_n)=\infty$. In 1973, Bondy proposed the metaconjecture that almost every nontrivial condition implying Hamiltonicity sho...
Cai-Li Jia, Xiang-Ge Liu, Lu Yong et al.· Electronic Journal of Combin...· 0 citations
It was recently shown that every connected graph of order $n \geq 5$ and size $m$ satisfies $\chi(G) \leq \left\lceil \frac{m}{\sqrt{n}} \right\rceil$, and it was asked whether the stronger inequality $\chi(G) \leq \left\lceil \frac{m}{2\sqrt{n}} \right\rceil + 1$ holds for every connected triangle-free graph. In this...
Let $G$ be a simple graph on $n$ vertices and $e(G)$ edges. Let $\mu_1\geq \cdots \geq \mu_{n-1}\geq \mu_n=0$ be the Laplacian eigenvalues of $G$. For $k=1, \ldots, n$, let $S_k(G)=\sum_{i=1}^{k}\mu_i$. Brouwers conjecture asserts that for any $k\in\{1,\ldots, n\}$, $S_k(G)\leq e(G)+\binom{k+1}{2}$. In [Bounding the su...
Let $G$ be a graph on $n$ vertices, and let $e_G(S,V(G)\setminus S)$ be the number of edges with exactly one endpoint in $S$. The Cheeger constant and the restricted Cheeger constant of $G$, where $k\ge1$ is real, are, respectively, \[ h(G)=\min_{\substack{\emptyset\ne S\subseteq V(G)\\|S|\le \frac{n}{2}}} \frac{e_G(S,...
Cheng-Li Li, Le Xu, Bo Zhou· 1 citation· ⚡1
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