| The graph coloring problem is an important branch in graph theory.The subject of graph theory originated from the famous K?nigsberg Seven Bridges problem.The vertex coloring problem of graph theory is a hot topic and it originated from ”four-color conjecture” proposed by Guthrie in 1852.With the development of graph theory,Vizing proposed the list coloring,which is a generalization of the proper coloring.In 2015,Dvo?ák and Postle proposed the DPcoloring,which is a generalization of the list coloring.Before and during this period,some other colorings were proposed successively,such as the vertex arboricity and the degenerate problems,etc.In 2018,T.Wang proposed the problem of strictly f-degenerate transversal of graphs(SFDT for short),which is a generalization of all above colorings and this has a great research significance.By studying the related papers of vertex arboricity,DP-coloring and degenerate problems,we can better study the SFDT problem of graphs.Whether there has an SFDT in G is a problem worthy studying.In this paper,it is divided into five chapters.The first two chapters are about some basic terms and definitions,as well as some related conclusions.In chapter three,we give a result on strictly f-degenerate transversal of toroidal graphs without small subgraphs,which improves many known results on(list)vertex arboricity and DP-coloring.In chapter four,we present a result on strictly f-degenerate transversal of planar graphs without intersecting 5-cycles,which improves that every planar graph without intersecting 5-cycles is DP-4-colorable.Chapter five is the summary and prospect. |
