arXiv subjects

Jia-Bao Yang

Publications and source records attributed to Jia-Bao Yang.

4 recordsLinked to original sources

arXiv2026

Further Results on the Maximum Number of Stars in Graphs with Forbidden Properties

A graph $G$ is called $k$-edge-hamiltonian if every linear forest (i.e., a disjoint union of paths) with at most $k$ edges is contained in a Hamilton cycle of $G$. In 2018, Füredi, Kostochka and Luo determined the maximum number of $t$-stars in nonhamiltonian graphs, thereby extending an earlier result of Erdős. Recently, Berikkyzy, Hogenson, Kirsch and McDonald extended this line of research by determining the maximum number of $t$-stars in graphs that are not $k$-edge-hamiltonian, as well as in graphs failing to satisfy related properties such as traceability, hamiltonian-connectedness and $k$-hamiltonicity. For sufficiently large $t$, they also characterized the extremal graphs, while for smaller values of $t$, they proposed a conjecture. In this paper, we investigate this conjecture. We show that the conjecture fails at the critical value and further establish a threshold-type result describing the behavior of the extremal graphs when $t$ is close to this critical value.

math.CO↗

arXiv2026

Counterexamples to the Balogh-Linz-Patkós Conjecture

A set system $\mathcal{F}$ is called $t$-intersecting if $|A\cap B|\ge t$ for every pair of sets $A,B\in \mathcal{F}.$ A set system $\mathcal{F}$ is $k$-Sperner if it does not contain a chain of length $k+1$. Balogh, Linz and Patkós (Combinatorial Theory, 2023) conjectured an extremal result for $t$-intersecting $k$-Sperner families when $n+t$ is odd. In this note we give an explicit construction that is $t$-intersecting and $k$-Sperner, and whose size exceeds that of the conjectured fixed-star construction for infinitely many values of $n$. Consequently, we disprove the Balogh-Linz-Patkós conjecture for all $t\ge 2$ and $k\ge 2$ satisfying $k(t-1)\ge t+1$.

math.CO↗

arXiv2026

The maximum number of triangles in graphs without vertex disjoint friendship graphs

Given graphs $H$ and $F$, the generalized Turán number $\mathrm{ex}(n,H,F)$ is the maximum number of copies of $H$ among all $n$-vertex $F$-free graphs. The friendship graph $F_k$ consists of $k$ triangles sharing a common vertex. In this paper, we determine the value of $\mathrm{ex}(n,K_3,(t+1)F_k)$, where $K_3$ is a triangle, $t\geq 1$ is an integer, and $(t+1)F_k$ denotes a union of $(t+1)$ pairwise vertex-disjoint copies of $F_k$. Moreover, we characterize the extremal structure. Our result can be viewed as a generalization of the result of Zhu, Chen, Gerbner, Győri, and Hama Karim, as well as of the remaining case left open by Wang, Ni, Liu, and Kang. In contrast to the extremal graphs of $F_k$, the extremal graphs of $(t+1)F_k$ undergo a fundamental change. This structure is also different from those of previous similar problems.

math.CO↗

arXiv2026

Stability results for Berge-matching in hypergraphs

Given a graph $F$, a hypergraph is called a Berge-$F$ if it can be obtained by expanding each edge of $F$ into a hyperedge containing it. Let $M_{k}$ denote the matching of size $k$. Kang, Ni, and Shan [12] determined the Turán number of Berge-$M_k$. Our main result shows that if an $r$-uniform hypergraph $H$ on $n$ vertices has nearly as many edges as the extremal in their theorem without containing $M_k$, then $H$ must be structurally close to certain well-specified graphs. Meanwhile, our result also implies several stability results, such as the stability version of the well-known Erdős-Gallai theorem (Erdős and Gallai, 1959 [5]).

math.CO↗