Degree-balanced decompositions of cubic graphs
Borut Lužar, Jakub Przybyło, Roman Soták
TL;DR
The paper addresses the problem of extracting highly irregular spanning subgraphs from cubic graphs by introducing an explicit, constructive edge-coloring method that realizes prescribed degree counts. The authors prove that every cubic graph on $n$ vertices, except for a small set of exceptions ($K_4$, $K_{3,3}$, $3K_4$), contains a spanning subgraph $H$ with $m(H,k)$ equal to either $\left\lfloor \frac{n}{4} \right\rfloor$ or $\left\lceil \frac{n}{4} \right\rceil$ for all $k\in\{0,1,2,3\}$, achieving deviations at most $1/2$ from $n/4$. The method is twofold: (i) a constructive connected-case decomposition into precise degree-count patterns via staged recoloring along a shortest cycle, and (ii) a careful extension to general (possibly disconnected) cubic graphs using component-wise decompositions and a minimal-counterexample argument. The results provide exact, tight bounds for cubic graphs and yield a complete confirmation of the conjecture of Alon and Wei in this graph class, with clear guidance on exceptional cases and potential generalizations to higher degrees. Overall, the work offers an explicit algorithmic framework for generating highly irregular spanning subgraphs and advances the understanding of degree-constrained subgraphs in regular graphs.
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.