arXiv subjects

Guangmiao Yu

Publications and source records attributed to Guangmiao Yu.

3 recordsLinked to original sources

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.

math.CO↗

arXiv2026

A note on long nontrivial cycle in Hamiltonian graphs

Let $G$ be an $n$-vertex graph containing a Hamiltonian cycle and with minimum degree at least $3$. Girão, Kittipassorn and Narayanan (Israel J. Math., 2019) proved that $G$ contains another cycle of length at least $n-O(n^{4/5})$. In this paper, we improve their bound to $n-O(n^{2/3})$. Our proof is combined with a constructive method, which is based on a poset result, and a nonconstructive method. And the bound is best possible under these two methods.

math.CO↗

arXiv2026

New results on proper orientation number of graphs

The proper orientation number $\vecχ(G)$ of an undirected graph $G$ is the minimum $k$ such that there exists an orientation of $G$ with all out-degrees at most $k$ and with different out-degrees for any two adjacent vertices. Chen, Mohar and Wu (JCTB, 2023) proved that if $G$ is a $r$-partite graph, then $\vecχ(G) \leq \frac{1}{2} \text{Mad}(G)+r^{1+o(1)}$, where $\text{Mad}(G)$ is the maximum average degree of $G$. Moreover, if $G$ is a bipartite graph, then $ \vecχ(G) \leq \lceil \frac{1}{2} \text{Mad}(G)\rceil +3$ and this bound is tight. They also asked whether $\vecχ(G)-\lceil \frac{1}{2} \text{Mad}(G)\rceil$ can be bounded by a linear function of $r$. In this paper, we first construct somewhat involved $r$-partite graphs with $\vecχ(G)\geq\lceil \frac{1}{2} \text{Mad}(G)\rceil +\lfloor\frac{5}{2}r\rfloor-2$, showing that a linear dependence on $r$ is unavoidable. We also prove that $ \vecχ(G) \leq\lceil \frac{1}{2} \text{Mad}(G)\rceil +7$ for every 3-partite graph $G$. This implies $\vecχ(G)\le 10$ for $3$-colorable planar graphs and $\vecχ(G)\le 9$ for outerplanar graphs, improving the corresponding bounds of Chen, Mohar, and Wu.

math.CO↗