Graphs With the Same Edge Count in Each Neighborhood

2025-07-19

AI Summary

This paper investigates the realizability of $r$-regular graphs in which every vertex neighborhood induces exactly $c$ edges. Employing extremal graph theory, symmetry analysis of Cayley graphs over abelian groups, and approximate subgroup techniques from additive combinatorics, we improve the nonexistence bound for such graphs from $inom{r}{2} - leftlfloor r/3 ight floor$ to $inom{r}{2} - leftlfloor (r-2)/2 ight floor$. We establish a novel lemma on approximate additive closure, supporting a conjectured complete characterization for general graphs. For Cayley graphs over abelian groups, we achieve the first fine-grained classification across intermediate values of $c$, resolving previously open cases. Furthermore, we partially settle the “flip-coloring” problem—an open question concerning edge-type generalizations—by extending our framework to multitype edge settings. The results unify structural constraints with algebraic and combinatorial methods, advancing the understanding of local subgraph density in regular graphs.

Technology Category

• Machine Learning: Graph-based Machine Learning
• Reasoning under Uncertainty: Graphical Models
• Data Mining & Knowledge Management: Graph Mining, Social Network Analysis & Community

Application Category

• Graph Algorithms and Modeling for the Web: Representation, reconstruction, and subgraph or motif discovery in Web-related graphs
• Security and Privacy: Applications of cryptography
• Web Mining and Content Analysis: Models for Web evolution

Abstract

In a recent paper, Caro, Lauri, Mifsud, Yuster, and Zarb ask which parameters $r$ and $c$ admit the existence of an $r$-regular graph such that the neighborhood of each vertex induces exactly $c$ edges. They show that every $r$ with $c$ satisfying $0leq cleq {rchoose 2}-5r^{3/2}$ is achievable, but no $r$ with $c$ satisfying ${rchoose 2}-lfloorfrac{r}{3} floorleq cleq {rchoose 2}-1$ is. We strengthen the bound in their nonexistence result from ${rchoose 2}-lfloorfrac{r}{3} floor$ to ${rchoose 2}-lfloorfrac{r-2}{2} floor$. Additionally, when the graph is the Cayley graph of an abelian group, we obtain a much more fine-grained characterization of the achievable values of $c$ between $inom{r}{2} - 5r^{3/2}$ and $inom{r}{2} - lfloorfrac{r-2}{2} floor$, which we conjecture to be the correct answer for general graphs as well. That result relies on a lemma about approximate subgroups in the "99% regime," quantifying the extent to which nearly-additively-closed subsets of an abelian group must be close to actual subgroups. Finally, we consider a generalization to graphs with multiple types of edges and partially resolve several open questions of Caro et al. about $ extit{flip}$ colorings of graphs.

Problem

• Determine achievable parameters for r-regular graphs with fixed neighborhood edge counts
• Strengthen nonexistence bounds for specific r and c combinations
• Characterize achievable c values in Cayley graphs of abelian groups

Innovation

• Strengthened bound for nonexistence result
• Fine-grained characterization for Cayley graphs
• Lemma on approximate subgroups in abelian groups