arXiv subjects

Hanzhi Bai

Publications and source records attributed to Hanzhi Bai.

5 recordsLinked to original sources

arXiv2026

Packing and Covering Cycles Through Prescribed Vertices

Let $G$ be a finite simple graph and let $S\subseteq V(G)$. We prove that the minimum number of vertices meeting every cycle that intersects $S$ is at most the maximum number of vertices of $S$ covered by a collection of vertex-disjoint cycles. This answers a question posed by Bowler, Ghorbani, Gut, Jacobs, and Reich [\emph{SIAM Journal on Discrete Mathematics} \textbf{40} (2026), 988--999]. An incidence-based reduction to their bidirected packing--covering theorem preserves the packing value and projects transversals without increasing their cardinality.

math.CO↗

arXiv2026

Hitting Maximum Independent Sets in Dense and Highly Connected Graphs

For a graph $G$, let $h(G)$ be the minimum cardinality of a vertex set meeting every maximum independent set of $G$. We establish two complementary reduction principles for the Bollob\'as--Erd\H{o}s--Tuza conjecture: the conjecture for arbitrary graphs is equivalent to its restriction to regular graphs of any fixed positive linear degree, and, within every hereditary graph class, a uniform sublinear bound is equivalent to a sublinear bound on graphs of every fixed positive linear vertex connectivity. We prove the sharp general estimate \[ h(G)\le \left\lfloor\frac{|V(G)|}{2\alpha(G)+\delta(G)-|V(G)|}\right\rfloor \] whenever the denominator is positive, with equality for balanced complete multipartite graphs. Consequently, every $3$-colorable graph of order $n$ with $\kappa(G)\ge\rho n$ and $\rho>1/3$ has a hitting set of size at most $\lfloor(\rho-1/3)^{-1}\rfloor$; direct use of a $3$-coloring improves this to $6$ when $\kappa(G)>4n/9$ and to the sharp bound $3$ when $\kappa(G)>n/2$. For dense regular graphs with independence ratio greater than $1/4$, we obtain a logarithmic bound, while constructions with linear degree and linear independence number show that $h(G)=\Omega(\sqrt n)$ can still occur. We also prove a logarithmic bound for near-regular $3$-colorable graphs and exhibit a critical family at connectivity $n/3$ that explains the limitations of the degree-surplus and degree-ratio methods.

math.CO↗

arXiv2026

Paths with Prescribed Endpoints in Semicomplete and Locally Semicomplete Digraphs

We study two open path problems with prescribed endpoints posed by Bang-Jensen and Gutin. The first asks for a longest $(x,y)$-path in a semicomplete digraph. The second asks whether a locally semicomplete digraph has a Hamiltonian $(x,y)$-path. For semicomplete digraphs, we solve the first problem when the endpoints lie in different strong components. We also prove that if a non-Hamiltonian longest $(x,y)$-path omits a set of vertices, then these vertices together with $x$ and $y$ have a Hamiltonian $(y,x)$-path. This gives an equivalent cycle problem. We then give an exact algorithm that runs in polynomial time when the number of omitted vertices is fixed. For locally semicomplete digraphs, we determine the possible endpoints in the connected nonstrong case. Known results then leave only strong, nonsemicomplete, non-$4$-strong digraphs unresolved. Every such digraph of order at least five has a strong vertex cut of size at most three. Two examples show that a spanning directed path together with a vertex-disjoint directed cycle is not sufficient, and that vertices of one strong component need not occur consecutively on a Hamiltonian path.

math.CO↗

arXiv2026

Polynomial Algorithms for Minimum Degree Partitions in Semicomplete Digraphs

A 2-partition of a digraph is a partition of its vertex set into two nonempty parts. Degree-constrained 2-partition problems are generally computationally difficult, even when the prescribed properties are expressed only in terms of minimum indegree, minimum outdegree, or minimum semidegree. Bang-Jensen and Christiansen~\cite{B-C} conjectured that the minimum-degree partition problems would be polynomial-time solvable on semicomplete digraphs when the degree thresholds are fixed, and Bang-Jensen and Gutin~\cite{B-G-Classes} posed the related Problems~2.8.15 and~2.8.16. We resolve this conjecture. More precisely, for every fixed pair of integers $k_1,k_2\ge 2$, we give deterministic polynomial-time algorithms that decide whether a given semicomplete digraph admits a $(\delta^+\geq k_1,\delta^-\geq k_2)$-partition, a $(\delta^+\geq k_1,\delta^0\geq k_2)$-partition, or a $(\delta^0\geq k_1,\delta^0\geq k_2)$-partition, and construct such a partition whenever one exists. Here, $\delta ^+,\delta ^-,\delta ^0$ represent the minimum out-, in-, semi-degree, respectively. The algorithms use small degree certificates, minimal cores, closure and protective-set arguments, and deterministic universal colorings with monotone recoloring, which develop a new method in partition algorithm construction.

cs.DM↗

arXiv2026

New Tower-Type Lower Bounds for Hypergraph Ramsey Numbers

The Ramsey number $r_k(s,m)$ is the smallest $N$ such that any red/blue coloring of the $k$-subsets of $[N]$ contains a red $s$-set or a blue $m$-set. For fixed $k$ and $s$, and for sufficiently large $m$, the tower growth rate is determined by the stepping-up lemma, but for $s=m=k+1$ the available stepping-up lemmas do not apply. Fox asked for estimates of $r_k(k+1,k+1)$. Pudl\'ak, R\"odl, and Wesley gave the first tower-type bound: $r_k(k+1,k+1)\ge s_3(\lfloor k/4\rfloor)\ge 4\operatorname{twr}_{\lfloor k/4\rfloor-4}(2)$, where $s_3(k)$ is the $3$-color shift number and $\operatorname{twr}_1(2)=2$, $\operatorname{twr}_{i+1}(2)=2^{\operatorname{twr}_i(2)}$. In this paper, for $k\ge 6$, we improve the lower bound to $r_k(k+1,k+1)> s_3\bigl(\lfloor k/2\rfloor-2\bigr)$ by overcoming an obstruction in their construction. In addition, we give an exact characterization of $s_3(k)$ and, for $k\ge 5$, obtain a new explicit lower bound $s_3(k)\ge(\operatorname{twr}_{k-2}(2))^2$, which improves the result of Pudl\'ak and R\"odl. Consequently, for $k\ge 14$, $r_k(k+1,k+1)>(\operatorname{twr}_{\lfloor k/2\rfloor-4}(2))^2$.

math.CO↗