Injective Edge Chromatic Index of Generalized Petersen Graphs

Authors：Hu, Xiaolan^*; Legass, Belayneh-Mengistu

Source：Bulletin of the Malaysian Mathematical Sciences Society, 2023, 46(1): 37.

DOI：10.1007/s40840-022-01442-6

• Injective edge coloring
• Injective edge chromatic index
• Generalized Petersen graph

An injective k-edge coloring of a graph G=(V(G), E(G))is a k-edge coloring phi of G such that phi(e(1)) &NOTEQUexpressionL; phi(e(3)) for any three consecutive edges e(1), e(2) and e(3) of apath or a 3-cycle. The injective edge chromatic index of G, denoted by chi '(i)(G), is the minimum k such that G has an injective k-edge coloring. In this paper, we consider the injective edge coloring of the generalized Petersen graph P(n,k). We show that chi '(i)(P(n,k)) <= 4 if n equivalent to 0 (mod4) and k equivalent to 1 (mod2); and chi '(i )(P(n,k)) <= 5 if n equivalent to 2 (mod 4) and k equivalent to 1 (mod 2). Moreover, chi '(i )(P(n,3)) <= 5, chi ' i (P(2k+1,k)) <= 5 and chi '(i) (P(2k+2,k)) <= 5.