On the Chromatic Number of Cycle Books Graph

Abstract

Graph coloring is a fundamental topic in graph theory, with various applications in scheduling, networking, and optimization problems. In this study, we investigate the chromatic number of the cycle books graph , a structured graph formed by attaching multiple cycles to a common path . We establish that the chromatic number of depends on the parity of . Specifically, we prove that if is even, the chromatic number is , while if is odd, the chromatic number is . These results provide a deeper understanding of coloring properties in book-like graphs and contribute to the broader study of chromatic numbers in structured graph families. The findings may be extended to other variations of book graphs and related topologies in future research.

Keywords

• chromatic number, cycle books graph, graph coloring, cycle, graphs