On The Adjacent Strong Edge Coloring of 1-Tree

Abstract:

      Let G(V,E) be a graph. A k -proper edge coloring f is called a k -adjacent strong edge coloring of G(V,E) iff every uv∈ E(G) satisfies f[u] ≠ f[v], where f[u] = (f(uw) |uw∈E(G) }, is called k -ASEC for short, and χ_as(G) = min{k | There exists a k-ASEC of G} is called the adjacent strong edge chromatic number of G. In this paper,we present a conjec- ture that for 2-connected graph G(V,E) (G(V,E) ≠ C₅),△ (G) ≤χ_as(G) ≤ △(G) + 2, and prove that for 1-tree graph with △(G)≥4 have △(G) ≤ χ_as(G) ≤ △(G)+1 and χ_as(G) = △(G)+1 iff E(G[V_△])≠φ,where V_△=(u|u∈V(G),d(u)=△(G)}.