On equitable coloring of grapes

Abstract

This thesis entitled ON EQUITABLE COLORING OF GRAPHS newlineconsists of seven chapters In Chapter 1 the origin and development of graph theory is given briefly Also the relation of graph theory with other branches of mathematics is described briefly The basic definition of graphs are given in detail newlineIn Chapter 2 a brief litrature review on graph coloring has been discussed newlineby describing the origin of chromatic graph theory application of graph colorings newlinebasic definition related to equitable coloring and literature survey of equitable coloring newlineIn Chapter 3 the equitable coloring of some corona products G H of two newlinegraphs G and H has been considered In particular it has been shown that deciding newlinethe colorability of G H is NP complete even if G is 4 regular and H is K Next newlineexact values or upper bounds on the equitable chromatic number G H are newlinegiven where G is equitably 4colorable graph and H is a complete graph a cycle or newlinea path has been found In Chapter 4 the equitable chromatic number of G W the sun let graphs S line graph of sun let graphs LS middle graph of sun let graphs MS total newlinegraph of sun let graphs TS the helm graphs H the line graph of helm graphs newlineLH middle graph of helm graphs MH total graph of helm graphs TH newlinethe gear graphs G the line graph of gear graphs LG the middle graph of gear newlinegraphs MG the total graph of gear graphs TG has been found newlineIn Chapter 5 the equitable chromatic number for the line graph of Kn del newlinegraphs LW central graph of Kn del graphs CW and corona product of newlineKn¨odel graphs W W the complete graph and its complement K newlineK the star graph and its complement K K and the corona product of two complete newlinegraphs K K have been found and proved In Chapter 6 interesting results of the equitable chromatic number for the central graph of star graph C K the central graph of complete bigraph C K the total graph of complete bigraph T K the total graph of path T P and the total graph of cycle T C are obtained newline newline