arXiv subjects

Jifu Lin

Publications and source records attributed to Jifu Lin.

13 recordsLinked to original sources

arXiv2026

Optimal path decompositions of sparse random graphs

Let $\operatorname{odd}(G)$ be the number of odd-degree vertices of a graph $G$. We prove that, if $p=o(1)$ and $np/\log n\to\infty$, then with probability tending to one the random graph $G(n,p)$ has an edge decomposition into exactly $\operatorname{odd}(G)/2$ paths. This confirms a conjecture of Glock, Kühn and Osthus from 2016. The proof combines a local pairing argument based on the lopsided Lovász local lemma with cycle decompositions of random graphs.

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arXiv2026

A sharp bound on the binding number of $K_k$-free graphs and the Brandt conjecture

The binding number of a graph is the smallest ratio $|N(S)|/|S|$ among all nonempty vertex sets $S$ whose neighborhood $N(S)$ is not the whole vertex set. We prove that every finite nonempty $K_k$-free graph has binding number at most $k-2$ for every $k\ge6$, confirming Brandt's conjecture; the bound is sharp for complete $(k-1)$-partite graphs with equal nonempty parts. We first prove the case $k=6$ and then extend it to all $k\ge6$ by induction.

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arXiv2026

Sharp linear Turán remainders for cycle and edge extremality

For a graph $G$, let $e(G)$ denote the number of edges of $G$, and let $c(G)$ be the number of distinct cycles in $G$. Morrison, Roberts and Scott asked whether, for every fixed graph $H$ and all large $n$, some $n$-vertex $H$-free graph maximizes both $e(G)$ and $c(G)$. In this paper, we show that the answer is no for every possible chromatic number. Let $T_{n,r}$ be the complete $r$-partite Turán graph on $n$ vertices. We find a constant $γ_r>0$ such that if $n$ is sufficiently large and $\mathcal{H}$ is a finite family with $χ(\mathcal{H})=r+1$ satisfying \[ ex(n,\mathcal{H})<e(T_{n,r})+γ_r n, \] then every cycle-maximal $\mathcal{H}$-free graph is edge-extremal, and $γ_r$ is best possible.

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arXiv2026

An asymptotically tight bound on oriented diameter

Let $f(d)$ be the smallest integer such that every finite connected bridgeless simple graph of diameter $d$ admits a strong orientation of diameter at most $f(d)$. Chvátal and Thomassen (1978) proved $\lceil d^2/2\rceil+d\le f(d)\le2d^2+2d$. In this paper, we prove that $\lceil d^2/2\rceil+d\le f(d)\le\lceil d^2/2\rceil+d+18$ for every $d\ge2$, which shows that $f(d)=\lceil d^2/2\rceil+d+O(1)$ and determines both the quadratic and linear terms up to a bounded additive error. The key method of our proof is to construct a central subgraph $H$ that admits a strong orientation of diameter $O(d)$ and is within distance $\lfloor d/2\rfloor$ of every vertex outside it. We obtain the sharp linear coefficient by jointly estimating outside paths and their connecting paths in $H$ at the actual attachment vertices, rebuilding $H$ when necessary.

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arXiv2026

An asymptotically tight bound on oriented diameter

Let $f(d)$ be the smallest integer such that every connected bridgeless graph of diameter $d$ admits an orientation with directed diameter at most $f(d)$. Chvátal and Thomassen (1978) proved $\left\lceil{d^2}/{2}\right\rceil+d\le f(d) \le 2d^2+2d$. In this paper, we prove that $f(d)\le \left\lceil d^2/2\right\rceil+6d$, which shows $f(d)= d^2/2+Θ(d)$ and determine the leading quadratic coefficient of $f(d)$. The key method of our proof is to construct a central subgraph $H$ such that it admits a strong orientation of diameter $O(d)$ and the distance between $H$ and every vertex outside $H$ is at most $ d/2 $.

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arXiv2026

The maximum number of cycles of a given length in a nonhamiltonian graph

In 2026, Li and Zhan characterized the nonhamiltonian graphs of order $n$ with the maximum number of paths of length $k$, where $n$ and $k$ are integers satisfying $1\leq k\leq n-1$. This work solves and generalizes a problem proposed by Erdős in 1980. In this paper, we further determine the nonhamiltonian graphs of order $n$ attaining the maximum number of cycles of length $k$ for given integers $n$ and $k$ with $3\leq k\leq n-1$. As a corollary, we determine the generalized Turán number $\mathrm{ex}(n,C_k,C_n)$ for every $3\leq k\leq n-1$.

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arXiv2026

Oriented diameter of graphs with diameter $4$ and given edge girth

Let $f(d)$ be the smallest value for which every bridgeless graph $G$ with diameter $d$ admits a strong orientation $\overrightarrow{G}$ such that the diameter of $\overrightarrow{G}$ is at most $f(d)$. Chvátal and Thomassen (JCT-B, 1978) obtained general bounds for $f(d)$ and proved that $f(2)=6$. Kwok et al. (JCT-B, 2010) proved that $9\leq f(3)\leq 11$. Wang and Chen (JCT-B, 2022) determined $f(3)=9$. Babu et al. (DAM, 2021) showed $f(4)\leq 21$. In this paper, we introduce a new approach to studying $f(d)$ via the edge girth of a bridgeless graph $G$, denoted by $g^*(G)=\max\{l_G(e)\mid e\in E(G)\}$, where $l_G(e)$ is the length of the shortest cycle containing $e$ in $G$. Then we define $F(d,g^*)=\max\{\overrightarrow{diam}(G)\mid G\text{ is bridgeless},d(G)=d,g^*(G)=g^*\}$, and show $f(d)=\max\{F(d,g^*)\mid 2\leq g^*\leq 2d+1\}$. As the main result of this paper, we establish $F(4,2)=4$, $F(4,9)=12$, $F(4,3)\le 12$, and $F(4,g^*)\le 13$ for $g^*\in\{6,7,8\}$, and we propose two open problems for further research.

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arXiv2026

An improved upper bound on the oriented diameter of graphs with diameter $4$

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arXiv2026

Size conditions for admissible or consecutive even cycles in graphs

In 2022, Gao, Huo, Liu, and Ma proved that every graph with minimum degree at least $k+1$ contains $k$ admissible cycles, where a set of $k$ cycles is said to be admissible if their lengths form an arithmetic progression with common difference one or two. In this paper, we provide a sharp size analogue of their result and characterize the extremal graphs attaining the lower bound. In 2016, Verstraëte conjectured that every $n$-vertex graph $G$ containing no $k$ cycles of consecutive even lengths has at most $(2k+1)(n-1)/2$ edges, with equality only if every block of $G$ is a clique of order $2k+1$. We prove this conjecture for $2k+2\leq n\leq 4k+1$, and in fact obtain a stronger result in this range.

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arXiv2026

Signed graphs with exactly two main eigenvalues: The unicyclic case

An eigenvalue $λ$ of a signed graph $S$ of order $n$ is called a main eigenvalue if its eigenspace is not orthogonal to the all-ones vector $j$. Characterizing signed graphs with exactly $k$ $(1\le k\le n)$ distinct main eigenvalues is a problem in algebraic and graph theory that has been studied since 2020. Du et al. (2024, 2026) characterized a class of signed graphs with exactly two main eigenvalues by analyzing a type of multigraph whose base graph is a tree. In this paper, we extend this study to the case where the associated multigraph has a unicyclic base graph, and we conclude by proposing several open problems.

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arXiv2025

Spectral conditions for spanning $k$-trees or $k$-ended-trees of $t$-connected graphs

Let $G$ be a connected graph of order $n$. A spanning $k$-tree of $G$ is a spanning tree with the maximum degree at most $k$, and a spanning $k$-ended-tree of $G$ is a spanning tree at most $k$ leaves, where $k\geq2$ is an integer. This paper establishes some spectral conditions for the existence of spanning $k$-trees or spanning $k$-ended-trees in $t$-connected graphs, which generalize the results of Fan et al. (2022) and Zhou (2010), and improve the results of Fiedler et al. (2010), Ao et al. (2023) and Ao et al. (2025).

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arXiv2025

The connectedness of friends-and-strangers graphs about graph parameters and others

Let $X$ and $Y$ be two graphs of order $n$. The friends-and-strangers graph $\textup{FS}(X,Y)$ of $X$ and $Y$ is a graph whose vertex set consists of all bijections $σ: V(X)\rightarrow V(Y)$, in which two bijections $σ$ and $ σ'$ are adjacent if and only if they agree on all but two adjacent vertices of $X$ such that the corresponding images are adjacent in $Y$. The most fundamental question about these friends-and-strangers graphs is whether they are connected. In this paper, we provide a sufficient condition regarding the maximum degree $Δ(X)$ and vertex connectivity $κ(Y)$ that ensures the graph $\textup{FS}(X,Y)$ is $s$-connected. As a corollary, we improve upon a result by Bangachev and partially confirm a conjecture he proposed. Furthermore, we completely characterize the connectedness of $\textup{FS}(X,Y)$, where $X\in\textup{DL}_{n-k,k}$.

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arXiv2025

The existence of a spanning tree with leaf distance at least $d$ and leaf degree at most $k$ via the size or the spectral radius with respect to the minimum degree

Let $k$, $d$ be a positive integer, $G$ be a connected graph of order $n$, $T$ be a tree. The leaf distance of a tree is defined as the minimum distance between any two leaves. For $v\in V(T)$, the leaf degree of $v$ in $T$ is the number of leaves adjacent to $v$, and the leaf degree of $T$ is defined as maximum leaf degree among the vertices of $T$. In this paper, motivated by the conjecture proposed by Kaneko (2001) and its subsequent partial confirmation by Erbes, Molla, Mousley and Santana (2017), we obtain lower bounds in terms of the size and the adjacent spectral radius to guarantee that $G$ contains a spanning tree with leaf distance at least $d$, where $4\leq d \leq n-1$. Furthermore, we obtain some tight conditions in $G$ for its size and spectral radius to ensure that $G$ has a spanning tree with leaf degree at most $k$, which improves the result of Ao, Liu, Yuan, Ng and Cheng (2023).

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