In this paper, all graphs are graphs without parallel edges and loop. Let Δ(G),δ(G) and dG(u,v) denote vertex set, edge set, maximum degree, minimum degree and the distance between u and v in G respectively.Call a graph G planar if it can be embedded in the Euclidean plane such that any two edges intersect only at their ends. Any such particular drawing of a planar graph is called a plane graph. For a plane graph G, we use F(G) to denote its face set. For ienotes the degree of x in G, simply d(x).The k-2-distance coloring of a graph G is a mapping uch that The 2-distance chromatic number χ2(G) is the smallest integer k such that G has a k-2-distance coloring.L(1,1)-labeling of G is a mapping uch that for every pair u Similarly,λp,q(G) is called the L(p, g)-labeling of G.A list assignment of G is a mapping L that assigns a list L(v) colors to each vertex v ∈ V(G). Given a list assignment L of G, a 2-distance coloring c of G is called a L-2-distance coloring if c(v) ∈ L(v) for each vertex v ∈ V(G). A graph G is fc-2-distance choosable if G has an L-2-distance coloring for any list assignment L with\L(v)\> k for each vertex v ∈ V(G). The 2-distance choice number of G, denoted by ch2(G), is the least integer k such that G is k-2-distance choosable.This master thesis, consisting of four chapters, focuses on 2-distance (list) coloring on the condition of without short cycles for graphs. In the first chapter, we introduce some definitions and give a brief survey of 2-distance (list) coloring. In the second chapter, we mainly discuss 2-distance coloring of a planar graph without 3,4,7-cycles. In the third chapter, we study the 2-distance coloring of planar graphs without short cycles. In the fourth chapter, we study some sufficient conditions for the L-2-distance coloring of a graph. |