Study Of The Properties Of Planar Graphs And Some Colorings Of Graphs

In this paper, four problems are studied: the number of lower-degree vertices of a planar graph, the four color conjecture of planar graphs under some special conditions, the uniquely list coloring of graphs, the chromatic polynomial of complete bipartite and complete r -partite graphs.We shall first study the connected graphs, and prove that a graph is connected if andonly if its degree sequence d = (d₁,d₂,...,d_n) satisfies the conditions: is even, and for every 1≤k≤n ,; We will also prove that a graph is a tree if andonly if its degree sequence satisfies the condition: ; In the base of this, we will prove that the Four ColorConjecture holds under some conditions.We will also study the uniquely 3 list colorable graphs and will find a necessary and sufficient condition of a uniquely 3 list colorable graph. We will also find a sort of planar graphs whose m number is equal to 4.We will also prove the chromatic polynomial of complete bipartite graphs and the chromatic polynomial of complete r -partite graphs and get a combinatory formula.