The Radio k-chromatic Number for the Corona of Arbitrary Graph and $K_1$

Abstract

For $k>0$, an integer, and a connected simple graph H, a radio k-coloring of H is a function h which assigns every vertex of H a non-negative integer so that for each duo of two separate vertices x and y of H, the absolute difference of color of the vertices is at least $1+k-d(y,x)$. For a radio k-coloring h, the span $rc_k(h)$ of h is the largest color allotted by it. The radio k-chromatic number, $rc_k(H)$, is $\min \{rc_k(h):h~\text {is a radio}~k\text {-coloring of}~H\}$. In this manuscript, for the radio k-chromatic number of the corona of any arbitrary graph H and $K_1$, we obtain an upper bound. Further, we derive a necessary condition for the lower bound to be exact. Furthermore, we corroborate that the given upper bound is sharp for radio k-chromatic number of $P_n\odot K_1$, where $P_n$ is a path with n vertices and n is odd.