ADJACENT VERTEX-DISTINGUISHING TOTAL COLORING OF GENERAL MYCIELSKI OF Pn AND Cn

In this paper, we obtain the adjacent strong edge chromatic numbers of some General Mycielski Mn(G) (n ≥ 1) of special graphs G. AMS Subject Classification: 05C15


Introduction
Definition 1. (see [1] Received: August 7, 2015 c 2015 Academic Publications, Ltd. url: www.acadpubl.eu§ Correspondence author Definition 3. (see [2]) For a graph G(V, E), uv ∈ E(G), if a proper k-edge coloring f satisfy C(u) = C(v), then f is called k-adjacent strong edge coloring of G, which is abbreviated k-ASEC of G, and Conjecture 1. (see [2]) For any connected graph G with |V (G)| ≥ 3 and In [3]- [7] is studied the adjacent strong edge chromatic number of some special graph.In [8], is studied the adjacent strong edge chromatic number of Mycielski graph of some graph.
In this paper, we will examine the adjacent strong edge chromatic number of general Mycielski graph of complete graph with odd order, path, star, fan, wheel.

Main Results
Lemma 1. (see [9]- [11]) For complete graph K p with order p, we have where χ ′ (G) denotes edge chromatic number of G.
Theorem 2. For complete graph K p p ≡ 1( mod 2) and p ≥ 3, we have Mark this coloring method with f 0 .

by Lemma 3, we can color the edges by
In f 0 ∪ f , the color j will disappear in set of edges color of vertex v ij with 5. Now, we prove χ ′ as (M n (P m )) ≤ 5. We only prove the existence of 5-ASEC of M n (P m ).Let when j ≡ 1, 2(mod 3), f (v 0j v 0(j+1) ) = j(mod 3); Where Obviously, for any edge uv of graph M n (P m ), C(u) = C(v).So f is a 5-ASEC of M n (P m ).
Theorem 4. For S m (m ≥ 3), then For ∆(M n (S m )) = 2m and the maximum degrees are not adjacent, then Now, we prove χ ′ as (M n (S m )) ≤ 2m.We only prove the existence of 2m-ASEC of M n (S m ).Let f be: Obviously, for any edge uv of graph For ∆(M n (F m )) = 2m and the maximum degrees are not adjacent, then Now, we prove χ ′ as (M n (F m )) ≤ 2m.We only prove the existence of 2m-ASEC of M n (F m ).Let f be: others. Where, Where others. Where Where