List Injective Coloring Of Planar Graph

In this thesis,we only consider finite and simple graph G=(V,E).A kcoloring of graph G is a mapping c:V(G)→｛1,2,…,k}.The coloring c is called injective if any two vertices have a common neighbor get distinct colors,that is,for any vertices u and v,c(u)≠c(v)if NG(u)∩ NG(v)≠?.The injective chromatic number χi(G)of G is the smallest integer k such that G has an injective k-coloring.A list assigment of G is a mapping L which assigns a color list L(v)to each vertex v ∈ V(G).Given a list assignment L of G,an injective coloring c of G is called an injective L-coloring if c(v)∈ L(v)for any v ∈ V(G).A graph G is injectively kchoosable if G has an injective L-coloring for any list assignment L with |L(v)|≥ k for each vertex v.The least k for which G is injectively k-choosable is the injective choosability number or injective list chromatic number of G,denoted by χil(G).The thesis consists of four chapters,which focus on the injective list chromatic number of planar graph with maximal degree and some short cycle restricted.In chapter 1,we introduce the background and significance of the research,give the relevant notation and related research results about injective coloring,and show the main results of the thesis.In chapter 2,we show that for a planar graph G without 3-cycles and intersecting 4-cycles,if Δ(G)≥ 22,then Xil(G)≤Δ(G)+4;if Δ(G)≥15,thenχil(G)≤Δ(G)+5.In chapter 3,we prove that for a planar graph G with disjoint 5--cycles,ifΔ(G)≥18,then χil(G)≤ Δ(G)+3;if Δ(G)≥12,then χil(G)≤Δ(G)+4.In chapter 4,we provide some further questions that can be researched.