The List 2-distance Coloring Of Sparse Graphs With ? = 6 Or 7

All graphs considered in this paper are finite simple graph.A coloring φ :V（G）→ {1,2,…,k} of G is 2-distance if any two vertices at 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 χ₂（G）.If every vertex v ∈ V（G）has its own set L（v）of admissible colors,where|L（v）| ≥ k,then we say that V（G）has a list L of size k.A graph G is said to be list 2-distance k-colorable if any list L of size k allows a 2-distance coloring （?） such that （?）（v）∈ L（v）whenever v ∈ V（G）.The least k for which G is list 2-distance k-colorable is the list 2-distance chromatic number of G,denoted by ch₂（G）.In 1977,Wegner proved that if G is a planar graph with ? = 3,then χ₂（G）≤ 8and posed the following conjecture in the same paper: For each planar graph,（1）χ₂（G）≤ 7 if ?（G）= 3;（2）χ₂（G）≤ ?（G）+ 5 if 4 ≤ ?（G）≤ 7;（3）χ₂（G）≤ [3/2?（G）] + 1 if ?（G）≥ 8.And this conjecture has not yet been fully proved.This paper consists three chapters and mainly study the list 2-distance coloring of sparse graphs with ? = 6,7.In the first chapter,we introduce some basic definitions and the research status of 2-distance coloring,and gives the results of this paper.In the second chapter,we mainly study the list 2-distance coloring of sparse graphs with ? = 6,proving that:If G with △ = 6 and mad（G）< 2 +17/20（resp.mad（G）< 2 +2/10）,then ch₂（G）≤ 11（resp.ch₂（G）≤ 12）.In the third chapter,we mainly study the list 2-distance coloring of sparse graphs with ? = 7,showing that: If G with △ = 7 and mad（G）< 2 +4/5（resp.mad（G）< 2 +2/10）,then ch₂（G）≤ 11（resp.ch₂（G）≤ 12）.