Perfect folding of graphs

E M El-Kholy,H Ahmed

doi:10.21608/djs.2019.139192

Abstract

In this paper we introduced the definition of perfect foldingof graphs and we proved that cycle graphs of even number ofedges can be perfectly folded while that of odd number ofedges can be perfectly folded to C3 . Also we proved thatwheel graphs of odd number of vertices can be perfectlyfolded to C3. Finally we proved that if G is a graph of n verticessuch that 2 < clique number =chromatic number=k < n ,then the graph can be perfectly folded to a clique of order k.

Highlights

Let G = (V, E) be a graph, where V is the set of its vertices and E is the set of its edges
Consider the cycle graph C4 where (C4) = W (C4) = 2, the graph folding f : C4 C4 defined by f {v1,v4} ={v3,v2} and f {ei}={e3}, i=1,2,4 is a perfect folding, see Fig.(4)
Let G=C5 and h: G → G be the graph folding defined by h{v5,v4}={v3, v1} and h{ei} ={e2}, i=3,4 is a perfect folding, see Fig.(5)

Summary

Introduction

Definition (2-2) A graph map f :G1 G2 is called a graph folding if and only if f maps vertices to vertices and edges to edges ,i.e., if (i) For each vertex v V (G1) , f(v) is a vertex in V(G2) . If the chromatic number (G) is equal to two, G can be perfectly folded. This follows from the fact that the chromatic number of a bipartite graph is equal to two, and it can be perfectly folded.

Results

Conclusion