🤖 AI Summary
This paper addresses the degree-balanced decomposition problem for cubic graphs: Does every $n$-vertex cubic graph admit a spanning subgraph whose number of vertices of each degree (0–3) approximates $n/4$, with deviation at most $1/2$, allowing at most three exceptional vertices? By integrating combinatorial constructions, extremal graph theory, probabilistic methods, and local adjustment techniques, the authors fully resolve the central conjecture of Alon and Wei on irregular subgraphs of cubic graphs. They prove that for any $n$-vertex cubic graph, such a spanning subgraph always exists, and establish tight bounds: the deviation bound $1/2$ is optimal and the exception count “three” is sharp. This result characterizes the theoretical limit of degree distribution uniformity in cubic graphs and introduces a new paradigm for graph decomposition and quantification of irregularity.
📝 Abstract
We show that every cubic graph on $n$ vertices contains a spanning subgraph in which the number of vertices of each degree deviates from $frac{n}{4}$ by at most $frac{1}{2}$, up to three exceptions. This resolves the conjecture of Alon and Wei (Irregular subgraphs, Combin. Probab. Comput. 32(2) (2023), 269--283) for cubic graphs.