1-Harmonious coloring of triangular snakes

Approximate and exact results for the harmonious chromatic number

“…The harmonious chromatic number for the central graph, middle graph, and total graph of some families of graphs was studied in various papers: prism graph by Mansuri et al [16]; flower graph, belt graph, rose graph and steering graph by Muthumari and Umamamheswari [20]; snake derived architecture by Selvi [24]; Jahangir graph by Selvi and Azhaguvel [23]; star graph by Rajam and Pauline [22], and double star graph by Vernold et al [26].…”

ON HARMONIOUS CHROMATIC NUMBER OF C(Tm,n), M(Tm,n), C(L(Tm,n) AND M(L(Tm,n))

“…A proper vertex coloring of a graph G is a function c : V (G) −→ {1, 2, , k} in which c(u) and c(v) are different for the adjacent vertices u and v and smallest number of colors are needed to color a graph G is called its chromatic number, and is often denoted χ(G). The Harmonious coloring [5,6,7,9] of a simple graph G is proper vertex coloring in which no any two edges share the same color and minimum number of colors are to be used for harmonious coloring is known as the harmonious chromatic number, denoted by χ H (G). For a graph G = (V, E), subdividing each edge of the given graph G exactly once and joining all the non-adjacent vertices of it is the Central graph [3,7] C(G) of G and the middle graph M (G) [8] is defined in such a way that the vertex set of M (G) is V (G) ∪ E(G) and two vertices x, y of M (G) are adjacent in M (G)) when one of the following holds: (i) x, y are in E(G) and x, y are adjacent in G. (ii) x is in V (G), y is in E(G), and x, y are incident in G and the line Graph [4] of a simple graph G, denoted by L(G) and defined in such a way that there exactly one vertex v(e) in L(G) for each edge e in G and for any two edges e and e in G, L(G) has an edge between v(e) and v(e ), if and only if e and e are incident with the same vertex in G. The (m, n)-tadpole graph [1,2,4]…”