On edge chromatic number related to local antimagic labeling of some trees

Abstract

Tree graph is one of the simplest the most important classes of graphs. A tree is a graph having no cycles. Graph H(V, E) consists of a finite nonempty set V called vertices and element of E is called edge. Edge coloring is an assignment of labels color to edges of graph subject to certain constrains. Local antimagic total edge labeling is defined a bijection g : V (H) ∪ E(H) → {1,2, 3, . . ., p + q}, where p and q are the number of vertices and edges, respectively. If for any two adjacent edges e ₁ and e ₂, w_t (e ₁) ≠ w_t (e ₂), where for e = xy ∈ H, w_t (e) = g(x) + g(xy) + g(y). The local antimagic total edge labeling induces a proper edge coloring of H if each edge e is assigned the color w_t (e). The local antimagic total edge chromatic number of H denoted by χ_late (H), is the minimum of colors needed to color the edges of graph H. In this paper we determine the local antimagic total edge chromatic number of some families of trees, namely double star, broom, firecracker, and centipede graph. The all results attain the lower bound.