Year: 2026
Type: preprint
Abstract: A graph is \emph{elementary} if its edges can be coloured with two colours so that the two edges of every induced path on three vertices receive different colours. Elementary graphs are one of the two basic classes in the Chvátal--Sbihi decomposition of claw-free perfect graphs; they include all line graphs of bipartite multigraphs and all complements of bipartite graphs. We prove that every elementary graph $G$ satisfies $\ch(G)=χ(G)=ω(G)$, confirming for this class the conjecture of Gravier and Maffray that claw-free graphs are chromatic-choosable. The proof combines the Maffray--Reed description of elementary graphs as augmentations of line graphs of bipartite multigraphs, Galvin's theorem, an extremal (edge-maximal) choice of counterexample, and a Hall-type lemma describing how list colourings of a complete graph minus a matching can fail.
Source: arXiv (Cornell University)
Institution: Amrita Vishwa Vidyapeetham
Language: English
Subfield: Computational Theory and Mathematics
Field: Computer Science
Domain: Physical Sciences
Open Access status: green
OpenAlex ID: W7221752428