Maximum Spread of Vertex Degrees in a Simple Graph

TL;DR

This work investigates how far vertex degrees can spread in a simple graph by analyzing the minimum number of vertex pairs with degree difference less than a fixed $k$, denoted $h_k(G)$. It introduces the extremal value $f(n,k)=\min_G h_k(G)$ and provides an explicit construction $G_{n,k}$ achieving $h_k(G)=f_0(n,k)$, together with an upper bound $f(n,k)\le f_0(n,k)$ where $f_0(n,k)$ has a closed form in terms of $n$ and $k$. A central conjecture posits $f(n,k)=f_0(n,k)$ for all $n>k$, with the claim verified in several regimes: $k\le 2$, $k<n\le 2k$, $k|n$, and when the remainder $n\bmod k$ satisfies $\ge 2k/3$. The proofs develop a constructive extremal graph and a suite of lemmas using degree-interval partitions, Jensen-type convexity arguments, and detailed case analyses to bound $h_k(G)$ from below in the proven cases, linking the problem to combinatorial graph-structure constraints and related probabilistic spread results.

Abstract

We consider the following problem: let $n>k$ be natural numbers, and let $G$ be a graph on $n$ vertices (undirected, without loops or multiple edges). Denote by $h_k(G)$ the number of unordered pairs of vertices in the graph $G$ whose degrees differ by less than $k$. We aim to determine the smallest possible value $f(n,k)$ of the quantity $h_k(G)$. Interest in this question is motivated by the fact that the bipartite analogue of the problem enabled S. Cichomski and F. Petrov to prove the Burdzy -- Pitman conjecture on the spread of independent coherent random variables. The problem has been solved under a number of restrictions on $n$ and $k$. A conjecture about the answer in the general case is also presented.

Theorems & Definitions (27)

• Lemma 1
• proof
• Theorem 1
• Remark
• Theorem 2
• Theorem 3
• Theorem 4
• Theorem 5
• Definition 1
• proof : Proof of Theorem 1.