Locating Chromatic Number for Corona Operation of Path and Cycle

Source abstract

The locating chromatic number (<em>lcn</em>) of a graph is a part of discrete mathematic research, there is no general theorem for determining the <em>lcn</em> of any graph. The corona operation of <em>P</em><sub>n</sub> and <em>C</em><sub>m</sub>, denoted by <em>P</em><sub>n</sub>⊙<em>C</em><sub>m</sub> is defined as the graph obtained by taking one copy of <em>P</em><sub>n</sub> and |<em>V</em>(<em>P</em><sub>n</sub>)| copies of <em>C</em><sub>m</sub> and then joining all the vertices of the <em>k</em><sup>th</sup>-copy of <em>C</em><sub>m</sub> with the <em>k</em><sup>th</sup>-vertex of <em>P</em><sub>n</sub>. In this paper, we discuss the <em>lcn</em> for the corona operation of path and cycle. The <em>lcn</em> of (<em>P</em><sub>n</sub>⊙<em>C</em><sub>3</sub>) is 5 for 3 ≤<em>n</em>< 7 and 6 for <em>n</em>≥7. Moreover, the <em>lcn</em> of (<em>P</em><sub>n</sub>⊙<em>C</em><sub>4</sub>) is 5 for 3≤<em>n</em>< 6 and 6 for <em>n</em>≥6