arXiv subjects

Jiangdong Ai

Publications and source records attributed to Jiangdong Ai.

At least 37 records · Page 2Linked to original sources

arXiv2026

A weighted cycle-localization inequality

In 1959, Erdős and Gallai showed that every $2$-connected graph $G$ contains a cycle of length at least $\frac{2|E(G)|}{|V(G)|-1}$. This result was subsequently extended to weighted graphs by Bondy and Fan in 1991. A natural local variant of this problem arises by considering, for each edge $e\in E(G)$, the quantity $c(e)$, defined as the length of the longest cycle in $G$ containing $e$ (with $c(e)=2$ if $e$ is a bridge). Zhao and Zhang recently proved that for every graph $G$ on $n$ vertices satisfies $\sum_{e\in E(G)}\frac{1}{c(e)}\le \frac{n-1}{2}.$ In this note, we establish a weighted generalization of this inequality. For a weighted graph $(G,w)$ with positive edge weights, let $C_w(e)$ denote the maximum weight of a cycle containing $e$ (setting $C_w(e)=2w(e)$ if $e$ is a bridge). We prove that $$ \sum_{e\in E(G)}\frac{w(e)}{C_w(e)}\le \frac{n-1}{2}. $$ Our result can be viewed as a weighted local analogue of the Bondy-Fan theorem, thereby establishing a correspondence between the global and local perspectives. Furthermore, we present a broad class of graphs attaining equality and derive necessary conditions for equality.

math.CO↗

arXiv2026

Refinements of Alon-Babai-Suzuki-type intersection theorems via non-shadows and binomial support

We prove a multilevel non-shadow refinement of the Alon--Babai--Suzuki (ABS) nonuniform restricted-intersection theorem. Let $K=\{k_1,\dots,k_r\}$ and let $L$ be a set with $|L|=s$. If $\mathcal{F}\subseteq \bigcup_{k\in K}\binom{[n]}{k}$ is $L$-intersecting and $k_i>s-r$ for every $i$, then $|\mathcal{F}| + \sum_{j=s-r+1}^{s} |\mathcal{N}_j(\mathcal{F})| \le N(n,s,r),$ equivalently $|\mathcal{F}| \le \sum_{j=s-r+1}^{s} |\partial_j\mathcal{F}|.$ Thus the ABS bound is sharpened by the total non-shadow deficit on the top $r$ levels. In the modular setting, we take a coefficient-sensitive viewpoint: the polynomial method depends not just on the degree of the annihilator polynomial $P_L(t)=\prod_{\ell\in L}(t-\ell)\in\mathbb{F}_p[t]$, but on which binomial terms actually appear in it. This yields a gap-free modular bound depending only on the active support levels of $P_L$. For almost-initial residue patterns $L=\{0,1,\dots,s-m-1\}\cup R \pmod p$ we obtain the collapse $|\mathcal{F}|\le \sum_{i=0}^{m}\binom{n}{s-i}.$ In particular, for consecutive residues $L=\{0,1,\dots,s-1\}\pmod p$ we get the sharp bound $|\mathcal{F}|\le \binom{n}{s}$, giving a partial negative answer to a question of Alon--Babai--Suzuki: the modular ABS bound $N(n,s,r)$ is not attainable in the consecutive-residue regime whenever $r\ge 2$.

math.CO↗

arXiv2026

Oriented discrepancy of Hamilton cycles in oriented graphs satisfying Ore-type condition

Erd{\H o}s (1963) initiated extensive graph discrepancy research on 2-edge-colored graphs. Gishboliner, Krivelevich, and Michaeli (2023) launched similar research on oriented graphs. They conjectured the following extension of Dirac's theorem: If $D$ is an oriented graph on $n \ge 3$ vertices with minimum degree $δ(D) \ge n/ 2$, then $D$ contains a Hamilton oriented cycle with at least $δ(D)$ arcs in the same direction. This conjecture was proved by Freschi and Lo (2024) who posed an open problem to extend their result to an Ore-type condition. We propose two conjectures for such extensions and prove results which provide support to the conjectures.

math.CO↗

arXiv2026

On a weaker notion of cross $t$-intersecting families

We prove that if two families $\mathcal{F} \subseteq \binom{[n]}{k}$ and $\mathcal{F}' \subseteq \binom{[n]}{k'}$ satisfy $\sum_{1 \leq i, j \leq \ell} \lvert F_i \cap F_j' \rvert \geq \ell^2t - \ell +1$ for every choice of distinct $F_1, \ldots, F_\ell \in \mathcal{F}$ and $F_1', \ldots, F_\ell' \in \mathcal{F}'$, then $\lvert \mathcal{F} \rvert \cdot \lvert \mathcal{F}' \rvert \leq \binom{n-t}{k-t} \binom{n-t}{k'-t}$, provided that $n$ is sufficiently large. This extends a celebrated theorem of Pyber for large $n$, which determines the tight upper bound for the product of the sizes of cross $1$-intersecting families.

math.CO↗

arXiv2025

Feedback vertex sets of digraphs with bounded maximum degree

A digraph $D$ is an oriented graph if $D$ does not have a pair of opposite arcs. The degree of a vertex $v$ of $D$ is the sum of the in-degree and out-degree of $v.$ Let $fvs(D)$ be the minimum number of vertices whose deletion from $D$ makes it acyclic. Let $D$ be a digraph with $n$ vertices and maximum degree $Δ$. We prove the following bounds. If $D$ is an oriented graph, then $fvs(D)\leq \frac{3n}{7}$ when $Δ\le 4$ and $fvs(D)\leq \frac{n}{2}$ when $Δ\le 5$. If $D$ is a connected digraph, $Δ\le 4$ and $D$ is not obtained from an odd undirected cycle by replacing every edge with the pair of opposite arcs with the same endvertices, then $fvs(D)\leq \frac{n}{2}$. If $D$ is an arbitrary digraph with $Δ\le 5$ then $fvs(D)\leq \frac{2n}{3}.$ Note that all the above bounds are tight.

math.CO↗

arXiv2025

Odd Induced Subgraphs in Graphs of Maximum Degree Four

A graph is called odd if all of its vertex degrees are odd. A long-standing conjecture asked whether there exists a positive constant $c$ such that every $n$-vertex graph without isolated vertices contains an odd induced subgraph on at least $cn$ vertices. In 2022, Ferber and Krivelevich resolved this conjecture affirmatively with $c=10^{-4}$. A natural question is to determine the largest possible constant $c$. In 1994, Caro remarked that if $2/7$ is a valid value for $c$, then it is the largest possible one. To the best of our knowledge, the bound $c\ge 2/7$ has not been improved. Previous research has established tight bounds for specific graph classes -- for instance, $c = 2/5$ for graphs with maximum degree at most $3$ and without isolated vertices. In this paper, we prove that $c=2/7$ is the tight bound for graphs with maximum degree at most $4$ and without isolated vertices. Our result provides some support for $2/7$ being the largest value of $c$.

math.CO↗

arXiv2025

Note on Long Directed Cycles in Eulerian Digraphs

Huang, Ma, Shapira, Sudakov and Yuster (Comb. Prob. Comput. 2013) proved that every Eulerian digraph of average out-degree $d$ has a directed cycle of length at least $\sqrt{d}.$ We improve the lower bound from $\sqrt{d}$ to $\sqrt{2d}-3/2.$

math.CO↗

arXiv2025

On high discrepancy $1$-factorizations of complete graphs

We proved that for every sufficiently large $n$, the complete graph $K_{2n}$ with an arbitrary edge signing $σ: E(K_{2n}) \to \{-1, +1\}$ admits a high discrepancy $1$-factor decomposition. That is, there exists a universal constant $c > 0$ such that every edge-signed $K_{2n}$ has a perfect matching decomposition $\{ψ_1, \ldots, ψ_{2n-1}\}$, where for each perfect matching $ψ_i$, the discrepancy $\lvert \frac{1}{n} \sum_{e\in E(ψ_i)} σ(e) \rvert$ is at least $c$.

math.CO↗

arXiv2025

Strong Ramsey game on two boards

The strong Ramsey game $R(\mathcal{B}, H)$ is a two-player game played on a graph $\mathcal{B}$, referred to as the board, with a target graph $H$. In this game, two players, $P_1$ and $P_2$, alternately claim unclaimed edges of $\mathcal{B}$, starting with $P_1$. The goal is to claim a subgraph isomorphic to $H$, with the first player achieving this declared the winner. A fundamental open question, persisting for over three decades, asks whether there exists a graph $H$ such that in the game $R(K_n, H)$, $P_1$ does not have a winning strategy in a bounded number of moves as $n \to \infty$. In this paper, we shift the focus to the variant $R(K_n \sqcup K_n, H)$, introduced by David, Hartarsky, and Tiba, where the board $K_n \sqcup K_n$ consists of two disjoint copies of $K_n$. We prove that there exist infinitely many graphs $H$ such that $P_1$ cannot win in $R(K_n \sqcup K_n, H)$ within a bounded number of moves through a concise proof. This perhaps provides evidence for the existence of examples to the above longstanding open problem.

math.CO↗

arXiv2024

Spanning weakly even trees of graphs

Let $G$ be a graph (with multiple edges allowed) and let $T$ be a tree in $G$. We say that $T$ is $\textit{even}$ if every leaf of $T$ belongs to the same part of the bipartition of $T$, and that $T$ is $\textit{weakly even}$ if every leaf of $T$ that has maximum degree in $G$ belongs to the same part of the bipartition of $T$. We confirm two recent conjectures of Jackson and Yoshimoto by showing that every connected graph that is not a regular bipartite graph has a spanning weakly even tree.

math.CO↗

arXiv2024

Arc-disjoint in- and out-branchings in semicomplete split digraphs

An \emph{out-tree (in-tree)} is an oriented tree where every vertex except one, called the \emph{root}, has in-degree (out-degree) one. An \emph{out-branching $B^+_u$ (in-branching $B^-_u$)} of a digraph $D$ is a spanning out-tree (in-tree) rooted at $u$. A \emph{good $(u,v)$-pair} in $D$ is a pair of branchings $B^+_u, B^-_v$ which are arc-disjoint. Thomassen proved that deciding whether a digraph has any good pair is NP-complete. A \emph{semicomplete split digraph} is a digraph where the vertex set is the disjoint union of two non-empty sets, $V_1$ and $V_2$, such that $V_1$ is an independent set, the subdigraph induced by $V_2$ is semicomplete, and every vertex in $V_1$ is adjacent to every vertex in $V_2$. In this paper, we prove that every $2$-arc-strong semicomplete split digraph $D$ contains a good $(u, v)$-pair for any choice of vertices $u, v$ of $D$, thereby confirming a conjecture by Bang-Jensen and Wang [Bang-Jensen and Wang, J. Graph Theory, 2024].

math.CO↗

arXiv2024

A short note on spanning even trees

We call a tree $T$ is \emph{even} if every pair of its leaves is joined by a path of even length. Jackson and Yoshimoto~[J. Graph Theory, 2024] conjectured that every $r$-regular nonbipartite connected graph $G$ has a spanning even tree. They verified this conjecture for the case when $G$ has a $2$-factor. In this paper, we prove that the conjecture holds when $r$ is odd, thereby resolving the only remaining unsolved case for this conjecture.

math.CO↗

arXiv2024

A complete characterization of split digraphs with a strong arc decomposition

A \textbf{strong arc decomposition} of a (multi-)digraph $D(V, A)$ is a partition of its arc set $A$ into two disjoint arc sets $A_1$ and $A_2$ such that both of the spanning subdigraphs $D(V, A_1)$ and $D(V, A_2)$ are strong. In this paper, we fully characterize all split digraphs that do not have a strong decomposition. This resolves two problems proposed by Bang-Jensen and Wang and contributes to a series of efforts aimed at addressing this problem for specific graph classes. This work continues the research on semicomplete composition [Bang-Jensen, Gutin and Yeo, J. Graph Theory, 2020]; on locally semicomplete digraphs [Bang-Jensen and Huang, J. Combin. Theory Ser. B, 2010]; on a type of tournaments [Bang-Jensen and Yeo, Combinatorica, 2004].

math.CO↗

arXiv2024

Number of Subgraphs and Their Converses in Tournaments and New Digraph Polynomials

An oriented graph $D$ is converse invariant if, for any tournament $T$, the number of copies of $D$ in $T$ is equal to that of its converse $-D$. El Sahili and Ghazo Hanna [J. Graph Theory 102 (2023), 684-701] showed that any oriented graph $D$ with maximum degree at most 2 is converse invariant. They proposed a question: Can we characterize all converse invariant oriented graphs? In this paper, we introduce a digraph polynomial and employ it to give a necessary condition for an oriented graph to be converse invariant. This polynomial serves as a cornerstone in proving all the results presented in this paper. In particular, we characterize all orientations of trees with diameter at most 3 that are converse invariant. We also show that all orientations of regular graphs are not converse invariant if $D$ and $-D$ have different degree sequences. In addition, in contrast to the findings of El Sahili and Ghazo Hanna, we prove that every connected graph $G$ with maximum degree at least $3$, admits an orientation $D$ of $G$ such that $D$ is not converse invariant. We pose one conjecture.

math.CO↗

arXiv2024

A variable version of the quasi-kernel conjecture

A quasi-kernel of a digraph $D$ is an independent set $Q$ such that every vertex can reach $Q$ in at most two steps. A 48-year conjecture made by P.L. Erdős and Székely, denoted the small QK conjecture, says that every sink-free digraph contains a quasi-kernel of size at most $n/2$. Recently, Spiro posed the large QK conjecture, that every sink-free digraph contains a quasi-kernel $Q$ such that $|N^-[Q]|\geq n/2$, and showed that it follows from the small QK conjecture. In this paper, we establish that the large QK conjecture implies the small QK conjecture with a weaker constant. We also show that the large QK conjecture is equivalent to a sharp version of it, answering affirmatively a question of Spiro. We formulate variable versions of these conjectures, which are still open in general. Not many digraphs are known to have quasi-kernels of size $(1-α)n$ or less. We show this for digraphs with bounded dichromatic number, by proving the stronger statement that every sink-free digraph contains a quasi-kernel of size at most $(1-1/k)n$, where $k$ is the digraph's kernel-perfect number.

math.CO↗

arXiv2024

On degree power sum in $P_k$-free graphs

Let $G$ be a graph on $n$ vertices with degree sequence $(d_1,d_2......d_n)$. For a real $p \geq 1$, let $D_p(G)=\sum_{i=1}^nd_i^p$. A Turán-type problem of degree power sum was initiated by Caro and Yuster \cite{caro2000degpower}: determining the function $D_p(n,H) :=\max \{D_p(G): \text{$G$ is an $n$-vertex $H$-free graph}\}$. They obtained some exact values for certain graphs $H$. For a path $P_k$, they mentioned that ``a close examination of the proof of Theorem 1.2 shows that the value of $n_0(k)$ in the statement of the theorem is $O(k^2)$", namely, they could show the $n$-vertex $P_k$-free graph with maximum degree power sum is $W_{n,k-1,\lfloor \frac{k}{2} \rfloor -1} = K_{\lfloor \frac{k}{2} \rfloor -1} \vee \left((n - \lceil \frac{k}{2} \rceil)K_1 \cup K_{1+k-2\lfloor \frac{k}{2} \rfloor} \right)$ when $n \geq c k^2$ for some constant $c$. In this note, we improve their result to a linear size of $k$ by a different approach. The bound is tight up to a constant factor.

math.CO↗

arXiv2024

Piercing independent sets in graphs without large induced matching

Given a graph $G$, denote by $h(G)$ the smallest size of a subset of $V(G)$ which intersects every maximum independent set of $G$. We prove that any graph $G$ without induced matching of size $t$ satisfies $h(G)\le ω(G)^{3t-3+o(1)}$. This resolves a conjecture of Hajebi, Li and Spirkl (Hitting all maximum stable sets in $P_{5}$-free graphs, JCTB 2024).

math.CO↗

arXiv2024

Solution on strong partition of $2$-balanced regular multipartite tournaments

We call a partition of a $c$-partite tournament into tournaments of order $c$ is strong if each tournament is strongly connected. The strong partition number denoted as $ST(r)$, represents the minimum integer $c'$ such that every regular $r$-balanced $c$-partite tournament has a strong partition with $c\geq c'$. Figueroa, Montellano-Ballesteros and Olsen showed the existence of $ST(r)$ for all $r\geq 2$ and proved that $5\leq ST(2)\leq 7$. In this note, we establish that $ST(2)=6$ and we also show the unique $2$-balanced $5$-partite tournament which has no strong partition.

math.CO↗