On the total vertex irregularity strength of comb product of cycle and path with order 3

Let G = ( V ( G ), E ( G )) be a graph and k be a positive integer. A total k -labeling of G is a map f: V ( G ) ∪ E ( G ) → {1, 2,…, k }. The vertex weight v under the labeling f is denoted by wf ( v ) and defined by wf ( v ) = f ( v ) + ∑ υν∈E(G)f ( ν∈ ). A total k -labeling of G is called vertex irregular if there are no two vertices with the same weight. The total vertex irregularity strength of G , denoted by tvs ( G ), is the minimum k such that G has a vertex irregular total k -labeling. Let G and H be two connected graphs. Let o be a vertex of H . The comb product between G and H , denoted by G ⊳ o H , is a graph obtained by taking one copy of G and | V ( G )| copies of H and grafting the i -th copy of H at the vertex o to the i -th vertex of G . In this paper, we determine the total vertex irregularity strength of comb product of cycle and path with order 3.