Abstract
In this paper, we prove the existence of the adjacent vertex distinguishing total coloringnof quadrilateral snake, double quadrilateral snake, alternate quadrilateral snake and double alternate quadrilateral snake in detail. Also, we present an algorithm to obtain the adjacent vertex distinguishing total coloring of these quadrilateral graph family. The minimum number of colors required to give an adjacent vertex distinguishing total coloring (abbreviated as AVDTC) to the graph G is denoted by avt(G).
Highlights
2 Preliminary Resultswe write some basic definitions and results which are needed for section.Definition 2.1 The Quadrilateral snake Qn is obtained from path Pn with v1, v2, vn vertices by replacing each edge of the path by C4 with a new vertices u1, u2, un 1 and w1, w2, wn 1 . n n 1V[Qn ] = vi i=1 i=17030 | P a g e February 2017 www.cirworld.comISSN 2347-1921 Volume 13 Number 01 Journal of Advances in Mathematics n 1E[Qn ] = (vivi 1 viui vi 1wi uiwi ) i=1Definition 2.2 The Alternate Quadrilateral snake A(Qn ) is obtained from the path Pn with every alternate edge of a path is replaced by C4
We write some basic definitions and results which are needed for section
The vertex set of Double Alternate Quadrilateral snake is given in the algorithm (4.2)
Summary
We write some basic definitions and results which are needed for section. Definition 2.1 The Quadrilateral snake Qn is obtained from path Pn with v1, v2, vn vertices by replacing each edge of the path by C4 with a new vertices u1, u2 , , un 1 and w1, w2 , , wn 1. Definition 2.2 The Alternate Quadrilateral snake A(Qn ) is obtained from the path Pn with every alternate edge of a path is replaced by C4. Definition 2.3 The Double Quadrilateral snake D(Qn ) consists of two quadrilateral snakes that have a common Path using the new vertices ui , wi , ui and wi for i = 1,2, , n 1. We present an algorithm to obtain the adjacent vertex distinguishing total chromatic number of quadrilateral snake and alternate quadrilateral snake and we discussed their color classes. Algorithm 3.1 Procedure: Adjacent vertex distinguishing total coloring of Quadrilateral snake Q(n) , for n 4. The edge set of A(Qn ) is given by n 1 vi vi
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