Radio and Radial Radio Numbers of Certain Sunflower Extended Graphs

Abstract

Communication systems including AM and FM radio stations transmitting signals are capable of generating interference due to unwanted radio frequency signals. To avoid such interferences and maximize the number of channels for a predefined spectrum bandwidth, the radio-k-chromatic number problem is introduced. Let$G=\left(V,E\right)$be a connected graph with diameter$d$and radius$\rho$. For any integer$k$,$1\le k\le d$, radio$k-$coloring of G is an assignment$\phi$of color (positive integer) to the vertices of$G$such that$d\left(a,b\right)+\left|\phi \left(a\right)-\phi \left(b\right)\right|\ge 1+k$,$\forall a,b\in V\left(G\right),$where$d\left(a,b\right)$is the distance between$a$and$b$in G. The biggest natural number in the range of$\phi$is called the radio$k-$chromatic number of G, and it is symbolized by$r_{ck}\left(\phi \right)$. The minimum number is taken over all such radio$k-$chromatic numbers of$\phi$which is called the radio$k-$chromatic number, denoted by$r_{ck}\left(G\right)$. For$k=d$and$k=\rho$, the radio$k-$chromatic numbers are termed as the radio number ($rn\left(G\right)$) and radial radio number ($rr\left(G\right)$) of$G$, respectively. In this research work, the relationship between the radio number and radial radio number is studied for any connected graph. Then, several sunflower extended graphs are defined, and the upper bounds of the radio number and radial radio number are investigated for these graphs.