Vertex Coloring Of Graphs With Disabled Structure

A coloring ?:V(G)?{1,2,…,k} of G is 2-distance if any two vertices' distance at most two from each other get different colors.The minimum number of colors in 2-distance coloring of G is its 2-distance chromatic number,denoted by?2(G).If each vertex v?V(G)has a set L(v)of admissible colors,then we say V(G)has a list L.A graph G is said to be 2-distance k-choosable if any list L of size k allows a 2-distance coloring ? such that ?(v)?(v)whenever v?V(G).The least k for which G is 2-distance k-choosable is the list 2-distance choice number of G.denoted by ?2l(G).As early as 1977,Wegner[17]proposed the famous conjecture of 2-distance coloring of planar graphs:for a planar graph G,?2(G)?7 if?(G)=3,?2(G)??(G)+5 if 4??(G)?7,?2(G)?[3/2?(G)]+1 if ?(G)?8.This conjecture has not been completely solved.So far,many scholars have devoted themselves to the study of the upper bound of the 2-distance chromatic number of planar graphs.A k-L(2,l)-labelling of G is a function ?:V(G)?{0,1,2,…,k} such that|?(x)-?(y)|?2,if x is adjacent to y and |?(x)-?(y)|?1,if x and y have a common neighbor.The least k denoted by ?2,1(G)is the L(2,1)-labelling number.If each vertex v?V(G)has a set L(v)of admissible colors,then we say V(G)has a list L.A graph G is said to be L(2,1)-2-distance k-choosable if any list L of size k allows a k-L(2,1)-labelling ? such that ?(v)?L(v)whenever v?V(G).The least k for which G is L(2,1)-2-distance k-choosable is the list L(2,1)-2-distance choice number of G,denoted by ?2,1l(G).The L(2,1)-labelling problem was first proposed by Griggs and Yeh[13],they conjectured that for any graph G,?2,1(G)??2 if?(G)>2.Two cycles are said to intersect when they have at least one common vertex.In this paper,we have the following results:Theorem 1.2 For every planar graph with neither 3-cycles nor intersect 4-cycles and ?(G)?18,?2l(G)??(G)+8.Theorem 1.3 For every planar graph with neither 3-cycles nor intersect 4-cycles and ?(G)?28,?2,1 l(G)??(G)+12.Theorem 1.4 For every planar graph with maximum degree ?(G)?4 and g(G)?6,?2l(G)?10.Theorem 1.5 For every planar graph with maximum degree ?(G)?4 and g(G)?10,?2l(G)?7.