Borodin–Kostochka’s conjecture on (P5,C4)-free graphs

• Published in: Journal of applied mathematics & computing Vol. 65; no. 1-2; pp. 877 - 884
• Main Authors: Gupta, Uttam K., Pradhan, D.
• Format: Journal Article
• Language: English
• Published: Berlin/Heidelberg Springer Berlin Heidelberg 01.02.2021; Springer Nature B.V
• Subjects: Apexes; Applied mathematics; Computational Mathematics and Numerical Analysis; Graphs; Mathematical and Computational Engineering; Mathematics; Mathematics and Statistics; Mathematics of Computing; Original Research; Theorems; Theory of Computation; Coloring; Borodin–Kostochka’s conjecture; free graphs; 68Q25; Chromatic bound; 05C69
• ISSN: 1598-5865; 1865-2085
• DOI: 10.1007/s12190-020-01419-3

Brooks’ theorem states that for a graph G , if Δ ( G ) ≥ 3 , then χ ( G ) ≤ max { Δ ( G ) , ω ( G ) } . Borodin and Kostochka conjectured a result strengthening Brooks’ theorem, stated as, if Δ ( G ) ≥ 9 , then χ ( G ) ≤ max { Δ ( G ) - 1 , ω ( G ) } . This conjecture is still open for general graphs. In this paper, we show that the conjecture is true for graphs having no induced path on five vertices and no induced cycle on four vertices.