A(1,0)-relaxed Strong Edge List Coloring Of Planner Graphs

For a graph G,we use V(G),E(G)and F(G)to denote its vertex set,edge set and face set,respectively.Let s and t be two non-negative integers.We call G is(s,t)-relaxed strong edge k-colorable if there exists a mapping ?:E(G)? {1,2,..,k} such that there is at most s edges adjacent to e and t edges at distance two apart from e assigned the same color as e.The(s,t)-relaxed strong chromatic index,?'(s,t)(G),is the minimum number of integer k such that G admits an(s,t)-relaxed strong k-edge coloring.For each edge e ? E(G),we assign a list L(e)of possible colors,where L={L(e)| e ?E(G)}.If there exists an(s,t)-relaxed strong edge coloring ? of G such that ?(e)? L(e)for all e ? E(G),then G is said to be(s,t)-relaxed strong edge L-colorable.If for any list assignment with |L(e)|? k for all e ? E(G),G is(s,t)-relaxed strong edge L-colorable,then we say that G is(s,t)-relaxed strong edge k-choosable.The(s,t)-relaxed strong chromatic index,ch'(s,t)(G),is the smallest integer k such that G is strong edge k-choosable.The(s,t)-relaxed strong edge coloring was first investigated by He and Lin in 2017.In this paper,We consider the graph with maximum degree or minimum girth.In Chapter 1,we will first introduce some basic concepts appeared in the thesis.Then,we will give a brief introduction to the achievements of its related field.Finally we list some main results concerning this direction.In Chapter 2-4,we shall use structural analysis and coloring extending techniques to achieve our main results which are listed as follows.(1)Every graph G with maximum degree 3 is(1,0)-relaxed strong edge 8-choosable.(2)Every graph G with maximum degree 4 is(1,0)-relaxed strong edge 17-choosable.(3)Every graph G with girth at least 8 is(1,0)-relaxed strong edge(3?(G)-3)-choosable.