On the Rainbow Mean Indexes of Caterpillars

• Main Authors: Garciano, Agnes, Marcelo, Reginaldo M, Ruiz, Mari-Jo P, Tolentino, Mark Anthony C
• Format: text
• Published: Archīum Ateneo 2023
• Subjects: brooms; caterpillars; double brooms; double stars; rainbow mean coloring; rainbow mean index; Mathematics; Physical Sciences and Mathematics
• Online Access: https://archium.ateneo.edu/mathematics-faculty-pubs/247; https://thaijmath2.in.cmu.ac.th/index.php/thaijmath/article/view/1548/1511
• Institution: Ateneo De Manila University

Let G be a simple connected graph and c an edge coloring with colors that are positive integers. Given a vertex v of G, we define its chromatic mean, denoted by cm(v), as the average of the colors of the incident edges. If cm(v) is an integer for each v ∈ V (G) and distinct vertices have distinct chromatic means, then c is called a rainbow mean coloring. The maximum chromatic mean of a vertex in the coloring c is called the rainbow mean index of c and is denoted by rm(c). On the other hand, the rainbow mean index of G, denoted by rm(G), is the minimum value of rm(c) among all rainbow mean colorings c of G. In this paper, we determine the rainbow mean indexes of families of caterpillars, including brooms, and double brooms.