On the radio number for corona of paths and cycles

P. K. Niranjan, Srinivasa Rao Kola · AKCE International Journal of Graphs and Combinatorics · 2019

Radio 𝑘-coloring of graphs is one of the variations of frequency assignment problem. For a simple connected graph 𝐺 and a positive integer 𝑘⩽𝑑⁢𝑖⁢𝑎⁢𝑚⁡(𝐺), a radio 𝑘-coloring is an assignment 𝑓 of positive integers (colors) to the vertices of 𝐺 such that for every pair of distinct vertices 𝑢 and 𝑣 of 𝐺, the difference between their colors is at least 1+𝑘−𝑑⁡(𝑢,𝑣). The maximum color assigned by 𝑓 is called its span, denoted by 𝑟⁢𝑐𝑘⁡(𝑓). The radio 𝑘-chromatic number 𝑟⁢𝑐𝑘⁡(𝐺) of 𝐺 is min⁡{𝑟⁢𝑐𝑘⁡(𝑓):𝑓is a radio𝑘-coloring of𝐺}. If 𝑑 is the diameter of 𝐺, then a radio 𝑑-coloring is referred as a radio coloring and the radio 𝑑-chromatic number as the radio number, denoted by 𝑟⁢𝑛⁡(𝐺), of 𝐺. The corona 𝐺⊙𝐻 of two graphs 𝐺 and 𝐻 is the graph obtained by taking one copy of 𝐺 and |𝑉⁡(𝐺)| copies of 𝐻, and joining each and every vertex of the 𝑖th copy of 𝐻 with the 𝑖th vertex of 𝐺 by an edge. In this paper, for path 𝑃𝑛 and cycle 𝐶𝑚, 𝑚≥5, we determine 𝑟⁢𝑛⁢(𝑃𝑛⊙𝐶𝑚) when 𝑛 is even, and give an upper bound for the same when 𝑛 is odd. Also, for 𝑚≥4, we determine the radio number of 𝑃𝑛⊙𝑃𝑚 when 𝑛 is even, and give both upper and lower bounds for 𝑟⁢𝑛⁢(𝑃𝑛⊙𝑃𝑚) when 𝑛 is odd.

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