Explicit and logarithmically improved chromatic bounds for $(P_2\cup P_3)$-free graphs

Abstract

We prove that every \((P_2\cup P_3)\)-free graph \(G\), with \(k=ω(G)\), satisfies \[ χ(G)=O\!\left(k^3\frac{\ln\ln k}{\ln k}\right). \] Thus the class admits an \(o(k^3)\) binding function. We also prove the explicit bound \[ χ(G)\le \frac{k^3}{8}+\frac{5k^2}{4}-4k+8 \qquad(k\ge8), \] with a sharper formula for odd \(k\). A further refinement gives an explicit cubic bound with leading coefficient \(104/837<1/8\).

arXiv comment22 pages. Substantially rewritten and retitled, with a new $O(k^3\ln \ln k/\ln k)$ chromatic bound for the entire $(P_2\cup P_3)$-free class ($k=\omega (G)$) and sharper explicit cubic bounds. The earlier row estimates and the quadratic bound under a rank-gap condition are not reproduced here; see v3 for those results