arXiv subjects

Chuanshu Wu

Publications and source records attributed to Chuanshu Wu.

4 recordsLinked to original sources

arXiv2026

A sharp asymptotic bound for odd cycles in planar graphs

For graphs $G$ and $H$, let $\mathbf N(G,H)$ denote the number of unlabeled, not necessarily induced copies of $H$ in $G$, and let $\mathbf N_{\mathcal P}(n,H)$ be the maximum of $\mathbf N(G,H)$ over all $n$-vertex planar graphs $G$. We prove that, for every fixed integer $m\geq 3$, $$\mathbf N_{\mathcal P}(n,C_{2m+1})=2m\left(\frac{n}{m}\right)^m+O_m\!\left(n^{m-1/5}\right).$$ The proof uses a sharp weighted cycle--path inequality for edge probability measures on finite complete graphs. This strengthens a conjecture of Heath, Martin, and Wells and, together with their reduction lemma, yields the stated asymptotic formula.

math.CO↗

arXiv2026

A subquadratic bound for generalized Turán numbers of odd cycles

For a graph $H$ and a family of graphs $\mathcal F$, let $\text{ex}(n,H,\mathcal F)$ denote the maximum number of copies of $H$ in an $\mathcal F$-free graph on $n$ vertices. For every integer $i\ge 3$, let $C_i$ denote the cycle of length $i$. For $r\ge 3$, set $\mathscr {C}_r=\{C_3,C_4,\ldots,C_r\},$ and set $\mathscr {C}_2=\varnothing$. In this paper, we prove that, for all integers $l>k\ge 2$, $$ \text{ex}(n,C_{2k+1},\mathscr {C}_{2k}\cup\{C_{2l+1}\}) =O_{k,l} \left(n^{2-\frac{1}{k(k+1)(l-k)}}\ \ \right). $$ Together with the known upper bounds for the number of triangles in $C_{2l+1}$-free graphs, this confirms a conjecture of Gerbner, Győri, Methuku, and Vizer.

math.CO↗

arXiv2026

The maximum number of paths of even length in a planar graph

For graphs $G$ and $H$, let $N(G,H)$ be the number of unlabeled, not necessarily induced copies of $H$ in $G$, and let $f(n,H)$ be the maximum of $N(G,H)$ over all $n$-vertex planar graphs $G$. Ghosh, Győri, Martin, Paulos, Salia, Xiao and Zamora conjectured that, for every fixed integer $\ell\ge 2$, \[ f(n,P_{2\ell+1}) =4\ell\left(\frac{n}{\ell}\right)^{\ell+1}+O(n^\ell). \] We prove the conjecture, including the stated error term. Along the way, we also settle the Cox--Martin optimization conjecture.

math.CO↗

arXiv2026

Counterexamples to Clique Immersion Conjecture for Direct Products

Let $G$ and $H$ be graphs, and let $G\times H$ denote their direct product. For a graph $G$, let $\operatorname{im}(G)$ be the largest integer $t$ such that $G$ contains a $K_t$-immersion. Collins, Heenehan, and McDonald conjectured that if $\operatorname{im}(G)=t$ and $\operatorname{im}(H)=r$, then \[\operatorname{im}(G\times H)\ge (t-1)(r-1)+1.\] We disprove this conjecture by constructing an infinite family of connected bipartite counterexamples.

math.CO↗