arXiv subjects

Junran Yu

Publications and source records attributed to Junran Yu.

4 recordsLinked to original sources

arXiv2026

A Counterexample to the Alon-Krivelevich Conjecture and a New Lower Bound for Choosability of Bipartite Graphs

For an integer $k\ge 2$ and $r^2>2(2k-1)\ln (2k-1)$, we construct a bipartite graph $G$ whose two vertex classes are disjoint copies of the family of all $k$-subsets of a $(2k-1)$-element set, two vertices being adjacent when their sets intersect in at most $r$ elements. Assigning each vertex its corresponding subset as its color list, $G$ is not colorable, and thus its choice number $\ch(G)\ge k+1$. This implies that there are graphs with maximum degree $Δ$ and $$\ch(G)\ge\Bigl(\frac1{32}-o(1)\Bigr)\frac{(\lnΔ)^2}{(\ln\lnΔ)^3}.$$ When $k\to\infty$, consequently $\ch(G)/\lnΔ$ tends to infinity, which disproves the Alon--Krivelevich conjecture. In order to show the uncolorability, we use tools from probability (conditional product measure, nested coupling, and Doob martingale), which we write in an elementary language of weighted counting, and topology (Brouwer fixed-point theorem).

math.CO↗

arXiv2026

Sharp bounds and extremal structure of minimally strongly $k$-connected digraphs

Let $D$ be a finite simple minimally strongly $k$-connected digraph, where $k\ge 2$, and let $n=|V(D)|$. Mader proved in 1985 that $|A(D)|\le 2k(n-k)$ for all sufficiently large $n$, with equality if and only if $D\cong \overleftrightarrow{K}_{k,n-k}$ (JCTB 1985). We prove that this bound holds whenever $n\ge 3k-1$ and that the order threshold $3k-1$ is best possible. Moreover, if $n\ge 3k$, then equality holds if and only if $D\cong \overleftrightarrow{K}_{k,n-k}$. The threshold $3k$ is also best possible. These results resolve the conjecture of Mader in (JCTB 1985).

math.CO↗

arXiv2026

On Perfect Divisibility of Bull-Free Graphs Without Long Paths

A graph $G$ is {\em perfectly divisible} if, for every induced subgraph $H$ of $G$, $V(H)$ can be partitioned into $A$ and $B$ such that $H[A]$ is perfect and $ω(H[B])<ω(H)$. Chudnovsky and Sivaraman [J. Graph Theory \textbf{90} (2019) 54-60] proved that every ($P_5$, bull)-free graph is perfectly divisible, while Chen and Xu [Discrete Appl. Math. \textbf{372} (2025) 298-307] proved the same for ($P_7,C_5$, bull)-free graphs. We extend these results by proving that every ($P_8,C_5$, bull)-free graph is perfectly divisible and that, letting $F$ denote the Grötzsch graph, a ($P_6$, bull)-free graph is perfectly divisible if and only if it is $F$-free.

math.CO↗

arXiv2026

Perfect divisibility and perfect-Pollyanna in bull-free graphs

A graph $G$ is {\em perfectly divisible} if, for each induced subgraph $H$ of $G$, $V(H)$ can be partitioned into $A$ and $B$ such that $H[A]$ is perfect and $ω(H[B])<ω(H)$. A {\em bull} is a graph consisting of a triangle with two disjoint pendant edges. Hoàng [Discrete Math. 349 (2026) 114809] proposed four conjectures: 1. $P_5$-free graphs are perfectly divisible; 2. Odd hole-free graphs are perfectly divisible; 3. Even hole-free graphs are perfectly divisible; and 4. $4K_1$-free graphs are perfectly divisible. Karthick et al. [Electron. J. Combin. 29 (2022) P3.19] proposed a conjecture: Fork-free graphs are perfectly divisible. In this paper, we prove that all of five conjectures above hold for bull-free graphs. Our results also generalize some results of Chudnovsky and Sivaraman [J. Graph Theory 90 (2019) 54--60] and Karthick et al. [Electron. J. Combin. 29 (2022) P3.19]. We say that a class ${\cal C}$ is {\em perfect-Pollyanna} if ${\cal C}\cap {\cal G}$ is perfectly divisible for any hereditary class ${\cal G}$ in which each triangle-free graph is 3-colorable. Let $H\in\{\text{house, hammer, diamond}\}$. In this paper, we prove that the class of $(\text{bull}, H)$-free graphs is perfect-Pollyanna. Let ${\cal C}$ be the class of $(\text{bull}, H)$-free graphs. This implies that ${\cal C}\cap {\cal G}$ is perfectly divisible if and only if all of triangle-free graphs in ${\cal G}$ are perfectly divisible. As corollaries, we show that $(\text{bull},{\cal H})$-free graphs are perfectly divisible, where ${\cal H}$ is one of $\{P_{11},C_4\},\{P_{14},C_5,C_4\}$, and $\{P_{17},C_6,C_5,C_4\}$.

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