TY - JOUR

T1 - On the Study of Rainbow Antimagic Connection Number of Comb Product of Friendship Graph and Tree

AU - Septory, Brian Juned

AU - Susilowati, Liliek

AU - Dafik, Dafik

AU - Lokesha, Veerabhadraiah

N1 - Funding Information:
This research is funded by PUI-PT Combinatorics & Graph, CGANT UNEJ, DRTPM No. 0753/E.5/AL.05/2022.
Publisher Copyright:
© 2022 by the authors.

PY - 2023/1

Y1 - 2023/1

N2 - Given a graph G with vertex set (Formula presented.) and edge set (Formula presented.), for the bijective function (Formula presented.), the associated weight of an edge (Formula presented.) under f is (Formula presented.). If all edges have pairwise distinct weights, the function f is called an edge-antimagic vertex labeling. A path P in the vertex-labeled graph G is said to be a rainbow (Formula presented.) path if for every two edges (Formula presented.) it satisfies (Formula presented.). The function f is called a rainbow antimagic labeling of G if there exists a rainbow (Formula presented.) path for every two vertices (Formula presented.). We say that graph G admits a rainbow antimagic coloring when we assign each edge (Formula presented.) with the color of the edge weight (Formula presented.). The smallest number of colors induced from all edge weights of antimagic labeling is the rainbow antimagic connection number of G, denoted by (Formula presented.). This paper is intended to investigate non-symmetrical phenomena in the comb product of graphs by considering antimagic labeling and optimizing rainbow connection, called rainbow antimagic coloring. In this paper, we show the exact value of the rainbow antimagic connection number of the comb product of graph (Formula presented.), where (Formula presented.) is a friendship graph with order (Formula presented.) and (Formula presented.), where (Formula presented.) is the path graph of order m, (Formula presented.) is the star graph of order (Formula presented.), (Formula presented.) is the broom graph of order (Formula presented.) and (Formula presented.) is the double star graph of order (Formula presented.).

AB - Given a graph G with vertex set (Formula presented.) and edge set (Formula presented.), for the bijective function (Formula presented.), the associated weight of an edge (Formula presented.) under f is (Formula presented.). If all edges have pairwise distinct weights, the function f is called an edge-antimagic vertex labeling. A path P in the vertex-labeled graph G is said to be a rainbow (Formula presented.) path if for every two edges (Formula presented.) it satisfies (Formula presented.). The function f is called a rainbow antimagic labeling of G if there exists a rainbow (Formula presented.) path for every two vertices (Formula presented.). We say that graph G admits a rainbow antimagic coloring when we assign each edge (Formula presented.) with the color of the edge weight (Formula presented.). The smallest number of colors induced from all edge weights of antimagic labeling is the rainbow antimagic connection number of G, denoted by (Formula presented.). This paper is intended to investigate non-symmetrical phenomena in the comb product of graphs by considering antimagic labeling and optimizing rainbow connection, called rainbow antimagic coloring. In this paper, we show the exact value of the rainbow antimagic connection number of the comb product of graph (Formula presented.), where (Formula presented.) is a friendship graph with order (Formula presented.) and (Formula presented.), where (Formula presented.) is the path graph of order m, (Formula presented.) is the star graph of order (Formula presented.), (Formula presented.) is the broom graph of order (Formula presented.) and (Formula presented.) is the double star graph of order (Formula presented.).

KW - antimagic labeling

KW - comb product of graphs

KW - discrete mathematics

KW - graph theory

KW - rainbow antimagic connection number

KW - rainbow coloring

UR - http://www.scopus.com/inward/record.url?scp=85146817580&partnerID=8YFLogxK

U2 - 10.3390/sym15010012

DO - 10.3390/sym15010012

M3 - Article

AN - SCOPUS:85146817580

SN - 2073-8994

VL - 15

JO - Symmetry

JF - Symmetry

IS - 1

M1 - 12

ER -