Divisible subdivisions of graphs in subdivisions of complete graphs

TL;DR

This work advances the theory of divisible subdivisions in subdivisions of complete graphs by establishing sharp, general bounds for the $q$-divisible subdivision number $s_q(H)$ and its refined versions. It introduces a connector-based framework to enforce zero–weight subdivision paths and derives a universal bound $s_q(H)\le (2q-1)m+2n-1+4q$, with a tighter prime-order bound $s_p(H)\le rac{3p-1}{2}m-rac{p-1}{2}n+rac{p+1}{2}$ for connected $H$. The authors resolve the $q=2$ case for 5-degenerate graphs, proving $s_2(H)=n+m$, and completely determine $s_q(T)=nq-q+1$ for trees and $s_q(T,1)$ and $s_2(C,1)$ for trees and cycles, respectively, through intricate case analyses and inductive constructions that exploit zero-weight paths and combinatorial number theory tools. The results illuminate the behavior of modular subdivision constraints in sparse graph settings and raise natural questions about tightness and generalization to broader graph classes and subdivision parameters.

Abstract

Let $\mathbb{Z}_q$ denote the cyclic group of order $q$. A $\mathbb{Z}_q$-edge-weighted $K_f$ is the complete graph $K_f$ equipped with a weight function $ω: E(K_f) \to \mathbb{Z}_q$. A subdivision of a graph $H$ in a $\mathbb{Z}_q$-edge-weighted $K_f$ is called a $q$-divisible subdivision of $H$ if every subdivision path has weight congruent to zero modulo $q$. Let $q\ge 2$ be an integer and let $H$ be a graph with $n$ vertices and $m$ edges. Define $s_q(H)$ to be the smallest number $f$ such that every $\mathbb{Z}_q$-edge-weighted $K_{f}$ contains a $q$-divisible subdivision of $H$. Das, Draganić, and Steiner raised the following question (Problem 4.1 in [Tight bounds for divisible subdivisions, J. Combin. Theory, Ser. B 165 (2024) 1-19]): Given $q\in\mathbb{N}$ and a subcubic graph $H$ with $n$ vertices and $m$ edges, is it true $s_q(H)= m(q - 1) + n$? They also established the upper bound $s_q(H)\le 7mq+8n+14q$ for such a graph $H$. In this paper, we improve this bound by showing that $s_q(H)\le (2q - 1)m + 2n - 1 + 4q$, and establishing a sharper bound $s_p(H)\le \frac{3p - 1}{2}m - \frac{p - 1}{2}n + \frac{p + 1}{2}$ for prime $p$ and connected $H$. We resolve this problem in the case $q=2$ by proving that $s_2(H) = m + n$ for any 5-degenerate graph $H$, and in the case $q\ge 2$ and $T$ being a tree, by showing that $s_q(T) = nq - q + 1$. Let $s_q(H,t)$ be the minimum number $f$ such that every $\mathbb{Z}_q$-edge-weighted $K_f$ contains a $q$-divisible $t$-subdivision of $H$, where a $t$-subdivision of $H$ is a subdivision of $H$ such that each edge of $H$ is subdivided exactly $t$ times. We also prove that $s_2(H,1)= m + n$, where $H$ is a tree or a cycle on $n$ vertices with $m$ edges.

Figures (6)

• Figure 1: Four possible types of triangles within a $\mathbb{Z}_2$-edge-weighted graph.
• Figure 2: Vertex-sharing triangles.
• Figure 3: The structure used in Case $\delta(H) = 5$.
• Figure 4: The structure in the proof of Theorem \ref{['ts']} (1).
• Figure 5: The structure of Case 2.

Theorems & Definitions (58)

• Theorem 1.2: AlonK
• Theorem 1.4
• Theorem 1.5
• Theorem 1.6
• Theorem 1.7
• Theorem 1.8
• Definition 2.1: Efficient $t$-clique
• Definition 2.2: $t$-connector
• Definition 2.3: $S$-connector
• Theorem 2.5: The Cauchy-Davenport Theorem, see Daven