| A proper k-total-coloring of a graph G is a mapping φ:V(G)∪E{G)→{1,2,…,k},such that any two adjacent or incident elements in V(G)∪ E(G)receive different colors.Let Cφ[v]={φ(v)} ∪{φ(uv)|uv ∈ E(G)} be the set of colors assigned to a vertex v and those edges incident to v.φ is strict neighbor-distinguishing if |Cφ[u]\Cφ[v]|≥ 1 and |Cφ[v]\Cφ[u]|≥ 1 for each edge uv ∈ E(G).The strict neighbor-distinguishing total index,denoted by xsnt(G),of G is the minimum integer k such that G has a k-strictly neighbor-distinguishing total coloring.A proper k-edge-coloring of a graph G is a mapping φ:E{G)→{1,2,…,k} such that φ(c1)≠φ(c2)for any two adjacent edges c1 and c2 in E{G).The chromatic indexχ’(G)of G is the minimum integer k such that G is proper k-edge-colorable.A proper k-edge-coloring φ of G is called a strong k-edge-coloring if any two edges at distance at most two get distinct colors.The strong chromatic index,denoted by χ’s(G),of G is the minimum integer k such that G is strongly k-edge-colorable.By the definitions,it is straightforward to see that χ’s(G)≥ χ’(G)≥Δ.The dissertation consists of three chapters as follows.In Chapter 1,we collect some basic notation and give a detailed survey on the related fields.In Chapter 2,we focus on the strict neighbor-distinguishing total coloring of graphs.We prove that every graph G with Δ≤3 has χsnt(G)≤6,every graph G with Δ≤4 hasχsnt(G)≤ 8,Moreover,It is shown that every Halin graph or 2-connected outerplanar graph G has χsnt(G)=Δ+2.In Chapter 3,the strong edge coloring of two special graphs are investigated.We show that every K4-minor free graph G with Δ≥ 4 and without adjacent Δ-vertices satisfies χ’s≤3Δ-3,and every graph G with Δ=5 and without adjacent 5-vertices satisfies χ’s≤31. |
