The Lucky Choice Number For Planar Graph Of Large Girth

A lucky labeling f of a graph G is a mapping which assigns to each vertex v a positive integer f(v),so that any two adjacent vertices have distinct neighbor-sums,where the neighbor-sum of a vertex v is Sf(v)? ?u?NG(v)f(u).The lucky choice number of G,?l(G),is the smallest positive integer k such that for any k-list assignment L of G,G has a lucky labeling f with f(v)? L(v)for each vertex v.The concept of lucky number and lucky choice number was introduced by Grytczuk et al.[5]and has attracted much attention.Brandt,Diemunsch and Jahanbekam stud-ied the lucky choice number of planar graphs of large girth.They proved that planar graphs of girth at least 5,6,7 and 26 have lucky choice number at most 19,9,8 and 3,respectively[4].In this thesis,we strengthened their results and proved that planar graphs of girth at least 5,6,7 and 21 have lucky choice number at most 15,8,7 and 3,respectively.