The study of vertex critical graphs

• Main Authors: Chong-Ren Guo, 郭崇人
• Other Authors: Chin-Mei Kau Fu
• Format: Others
• Language: zh-TW
• Published: 2003
• Online Access: http://ndltd.ncl.edu.tw/handle/10359315226454565075

碩士 === 淡江大學 === 數學學系 === 91 === A graph G is an order pair (V , E),where V is a nonempty finite set, called vertex set, and E is a subset of two elements set of V, called edge set. The vertex coloring of G is a coloring on the vertices of G such that two adjacent vertices are colored different colors. The minimum number of colors to the vertex coloring of G is called the chromatic number of G, denoted by  (G). G is k-critical if  (G) = k and  (G \ v) k, for any vertex v of G. In this thesis, we list all the critical graphs with n vertices and n  7. From those critical graphs we conclude three different kinds critical graphs, two of them are 4-critical graphs and the other is (k  2)-critical 2k-regular graph. We have the following two 4-critical graphs: one is the graph join odd vertices of a cycle of length 4k2 and connecting even vertices of the cycle to an extra vertex, the other one is formed by two vertices and one odd cycle, where these two vertices connecting to the vertices of two paths which decompose the cycle respectively. Finally, we obtain that a sufficient condition of a 2k-regular circulant graph C(n , D) being (k  2)-critical is n≧k( k＋1 ), n 1(mod k＋1), gcd (n , a) = 1 and D = {a , 2a , 3a ,…, ka }(mod n).