{"about":{"site":"https://codewithpapers.app","non_affiliation":"Code with Papers and Syntology are not affiliated with, endorsed by, or sponsored by Papers with Code, Meta, or the pwc-archive mirror.","licence":"CC BY-SA 4.0","licence_url":"https://creativecommons.org/licenses/by-sa/4.0/legalcode","attribution":"https://codewithpapers.app/attribution","modified":"archive material modified by Syntology; see the attribution page"},"url":"/paper/a-note-on-bar-x-coloring-and-hat-a-coloring-4","title":"A note on $\\bar{X}$-coloring and $\\hat{A}$-coloring 4-regular graphs","arxiv_id":"2401.05510","date":"2024-01-10","proceeding":null,"authors":["Jorik Jooken"],"abstract":"Let $\\partial_H(u)$ be the set of edges incident with a vertex $u$ in the graph $H$. We say that a graph $G$ is $H$-colorable if there exist total functions $f : E(G) \\rightarrow E(H)$ and $g : V(G) \\rightarrow V(H)$ such that $f$ is a proper edge-coloring of $G$ and for each vertex $u \\in V(G)$ we have $f(\\partial_G(u))=\\partial_H(g(u))$. Let $\\bar{X}$ be the graph obtained by adding three parallel edges between two degree one vertices of the graph $K_{1,4}$. Let $\\hat{A}$ be the graph obtained by adding two pendant edges to two different vertices of a triangle and then adding two edges between the degree two vertex and the two adjacent degree three vertices. Malnegro and Ozeki [Discrete Math. 347(3):113844 (2024)] asked whether every 4-regular graph with an even number of vertices and an even cycle decomposition of size 3 admits an $\\bar{X}$-coloring or an $\\hat{A}$-coloring and whether every 2-connected planar 4-regular graph with an even number of vertices admits such a coloring. Additionally, they conjectured that for every 2-edge-connected simple cubic graph $G$ with an even number of edges, the line graph $L(G)$ is $\\bar{X}$-colorable. In this short note, we discuss two algorithms for deciding whether a graph $G$ is $H$-colorable. We give a negative answer to the two questions and disprove the conjecture by finding suitable graphs, as verified by two independent algorithms.","url_abs":"https://arxiv.org/abs/2401.05510v1","url_pdf":"https://arxiv.org/pdf/2401.05510v1.pdf","source":{"archive":"pwc-archive (Hugging Face), CC BY-SA 4.0","snapshot":"2025-07-28","licence_url":"https://creativecommons.org/licenses/by-sa/4.0/legalcode","row_kind":"links_only","authors_date_abstract":"arXiv metadata, CC0 1.0 (https://info.arxiv.org/help/license), from the Kaggle arXiv metadata snapshot of 2026-09-12"},"code_links":[{"paper_slug":"a-note-on-bar-x-coloring-and-hat-a-coloring-4","repo_url":"https://github.com/jorikjooken/hcoloring","is_official":1,"mentioned_in_paper":1,"mentioned_in_github":0,"framework":"none","reach":null}],"tasks":[],"methods":[],"datasets_introduced":[],"methods_introduced":[],"results":[],"syntology":{"syntology_url":null,"atlas_url":null,"mcp":null,"developers":"https://syntology.ai/developers"},"arxiv_metadata":null,"syntology_extracted_results":null}