Feiyu Nan | Searcharxiv

arXiv2026

A sharp Randi\'c bound for K\"onig--Egerv\'ary graphs and a conjecture of Aouchiche, Hansen, and Zheng

Let $\alpha'(G)$ be the matching number of a graph $G$, and let its Randi\'c index be $R(G)=\sum_{uv\in E(G)}(d(u)d(v))^{-1/2}$. In 2006, Aouchiche, Hansen, and Zheng conjectured that the maximum of $R(G)-\alpha'(G)$ over all $n$-vertex graphs is attained by the complete bipartite graph whose smaller part has $\lfloor\frac{n+4}{7}\rfloor$ vertices; the conjecture has remained open since then. In this paper, we prove that every $n$-vertex K\"onig--Egerv\'ary graph, and in particular every bipartite graph, satisfies \[ R(G)\le\sqrt{\alpha'(G)\left(n-\alpha'(G)\right)}, \] and we characterize the graphs attaining equality as the bipartite graphs all of whose components are semiregular with a common degree ratio. The K\"onig--Egerv\'ary hypothesis cannot be dropped, but the Berge--Tutte formula reduces the general case to it, and in this way we determine the maximum of $R(G)-\alpha'(G)$ for every $n\ge4$, together with all extremal graphs. The conjecture is therefore false, and it fails for infinitely many orders: the optimal part size is governed by the proportion $\frac{2-\sqrt2}{4}$ rather than by $\frac17$. The two proportions give asymptotic slopes differing by less than $3.7\cdot10^{-5}$, which is why a search over graphs of small order does not distinguish them. The equality statement fails as well, since the extremal graphs are not only the complete bipartite ones.

math.CO↗

arXiv2025

Majority Edge Colouring of Hypergraph

Motivated by recent work on majority edge-colourings of graphs, we initiate the study of the corresponding problem for hypergraphs. First, sharpening the probabilistic argument by a $KL$ large-deviation estimate, we obtain a sufficient minimum-degree condition of order $k^3\log(kr)$ with the sharp large-deviation constant $ I_k:=D\!\left(\frac1k\middle\|\frac1{k+1}\right)=\Theta(k^{-3}), $ where $D(\cdot\|\cdot)$ denotes the binary relative entropy. Our main constructive result shows that every hypergraph of rank at most $r$ and minimum degree at least $2rk^2$ admits a $1/k$-majority $(k+1)$-edge-colouring. The proof is based on a hypergraph extension of the key discrepancy lemma used in the graph case. We also show that the logarithmic dependence on the rank can be determined asymptotically. If $\mu_k(r)$ denotes the least minimum-degree threshold that guarantees a $1/k$-majority $(k+1)$-edge-colouring for all hypergraphs of rank at most $r$, then for every fixed $k\ge2$, $ \mu_k(r)=\frac{\log r}{I_k}+O_k(\log\log r). $ In particular, the correct logarithmic threshold is of order $k^3\log r$. Finally, we determine the correct order of the degree--colour trade-off. For integers $k\ge2$, $p\ge1$, and $r\ge2$, let $\nu_{k,p}(r)$ denote the least integer $q$ such that every hypergraph of rank at most $r$ and minimum degree at least $kp$ admits a $1/k$-majority $q$-edge-colouring. Then $ \nu_{k,p}(r)=\Theta_{k,p}(r^{1/p}). $ In particular, minimum degree at least $k^2-k$ guarantees a $1/k$-majority $O_k(r^{1/(k-1)})$-edge-colouring, and this exponent is best possible.

math.CO↗