Some Circular Edge Chromatic Number,

A (k,d)-edge coloring (k,d∈N,k≥2d)oi a graph G = (V, E) is an assignment c of colors {0,1,…, k - 1} to the edges of G such thatwhenever two edges e_i and e_j are adjacent.The circular chromatic indexχ'_c(G) is defined byχ'_c(G) = ini{k/d : Ghas a(k,d) -edge coloring}.The study of circular chromatic indexχ'_c(G) of a graph G, which is a refinement of its chromatic indexχ'(G) ,has been very active in the past decade. Many nice results are obtained.In this paper, we prove several properties and determine exact value ofχ'_c(G) for some classes of graphs and achieve following results:(1) We determine exact value ofχ'_c(G) for some critical graphs of small order;(2) The graphs K'_2,2 and K'_3,3 are obtained by subdividing one edge of K'_2,2,K'_3,3,then(3) If G^* is a chain (Fig.10(e)),thenχ'_c(G^*) = 3;(4) The tensor product G₁□G₂ of G₁ and G₂ . For m,n∈N, we prove C_2m+1□C_2n is of class one andχ'_c(C_2m+1□C_2n) = 4; C_2m+1□C_2n+1 is of class twoand...