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All Journal BAREKENG: Jurnal Ilmu Matematika dan Terapan Compiler
Jaya Santoso
IT DEL
Computer Science & IT Control & Systems Engineering Economics, Econometrics & Finance Energy Engineering Mathematics Mechanical Engineering Physics Transportation

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THE PARTITION DIMENSION OF CYCLE BOOKS GRAPH B_(m,n) WITH A COMMON PATH P_2 Santoso, Jaya; Darmaji, Darmaji
BAREKENG: Jurnal Ilmu Matematika dan Terapan Vol 19 No 2 (2025): BAREKENG: Journal of Mathematics and Its Application
Publisher : PATTIMURA UNIVERSITY

Show Abstract | Download Original | Original Source | Check in Google Scholar | DOI: 10.30598/barekengvol19iss2pp791-804

Abstract

Suppose is a connected graph with elements of a set of vertices denoted by and a subset of . The distance between and is the shortest distance to every vertex in . Let be a partition of , where each subset belongs to . The representation of a vertex with respect to is defined as the set of distances from to each vertex in . If each representation of each vertex of is different, then the partition is called the resolving partition of , and the partition dimension is the smallest integer such that has a resolving partition with members. In this research, we show the partition dimensions of the cycle books graph . Cycle books graph is a graph consisting of copies of a cycle with a common path . The partition dimension of the cycle books graph for and is shown.
On the Chromatic Number of Cycle Books Graph Santoso, Jaya
Compiler Vol 14, No 1 (2025): May
Publisher : Institut Teknologi Dirgantara Adisutjipto

Show Abstract | Download Original | Original Source | Check in Google Scholar | DOI: 10.28989/compiler.v14i1.2930

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.
Co-Authors Darmaji Darmaji