The present paper deals with the gracefulness of unconnected graph （jC_（4n））∪P_m,and proves the following result：for positive integers n,j and m with n≥1,j≥2,the unconnected graph（jC_（4n））∪P_m is a gracef...The present paper deals with the gracefulness of unconnected graph （jC_（4n））∪P_m,and proves the following result：for positive integers n,j and m with n≥1,j≥2,the unconnected graph（jC_（4n））∪P_m is a graceful graph for m=j-1 or m≥n＋j,where C_（4n） is a cycle with 4n vertexes,P_m is a path with m＋1 vertexes,and（jC_（4n））∪P_m denotes the disjoint union of j-C_（4n） and P_m.展开更多
Let G be a simple graph. A total coloring f of G is called E-total-coloring if no two adjacent vertices of G receive the same color and no edge of G receives the same color as one of its endpoints. For E-total-colorin...Let G be a simple graph. A total coloring f of G is called E-total-coloring if no two adjacent vertices of G receive the same color and no edge of G receives the same color as one of its endpoints. For E-total-coloring f of a graph G and any vertex u of G, let Cf （u） or C（u） denote the set of colors of vertex u and the edges incident to u. We call C（u） the color set of u. If C（u） ≠ C（v） for any two different vertices u and v of V（G）, then we say that f is a vertex-distinguishing E-total-coloring of G, or a VDET coloring of G for short. The minimum number of colors required for a VDET colorings of G is denoted by X^evt（G）, and it is called the VDET chromatic number of G. In this article, we will discuss vertex-distinguishing E-total colorings of the graphs mC3 and mC4.展开更多
讨论了m个阶为4的圈的点不交的并图m C 4的点可区别I-全染色和点可区别VI-全染色.通过构造以色集合和空集为元素的矩阵,利用色集合事先分配法及具体的染色给出了m C 4的最优点可区别I-全染色以及最优点可区别VI-全染色,进而确定了图m C ...讨论了m个阶为4的圈的点不交的并图m C 4的点可区别I-全染色和点可区别VI-全染色.通过构造以色集合和空集为元素的矩阵,利用色集合事先分配法及具体的染色给出了m C 4的最优点可区别I-全染色以及最优点可区别VI-全染色,进而确定了图m C 4的点可区别I-全色数及点可区别VI-全色数.结论表明点可区别I-全染色猜想和点可区别VI-全染色猜想对图m C 4成立.展开更多
文摘The present paper deals with the gracefulness of unconnected graph （jC_（4n））∪P_m,and proves the following result：for positive integers n,j and m with n≥1,j≥2,the unconnected graph（jC_（4n））∪P_m is a graceful graph for m=j-1 or m≥n＋j,where C_（4n） is a cycle with 4n vertexes,P_m is a path with m＋1 vertexes,and（jC_（4n））∪P_m denotes the disjoint union of j-C_（4n） and P_m.
基金Supported by the National Natural Science Foundation of China (Grant No.10771091)the Scientific Research Project of Northwest Normal University (Grant No.NWNU-KJCXGC-03-61)
文摘Let G be a simple graph. A total coloring f of G is called E-total-coloring if no two adjacent vertices of G receive the same color and no edge of G receives the same color as one of its endpoints. For E-total-coloring f of a graph G and any vertex u of G, let Cf （u） or C（u） denote the set of colors of vertex u and the edges incident to u. We call C（u） the color set of u. If C（u） ≠ C（v） for any two different vertices u and v of V（G）, then we say that f is a vertex-distinguishing E-total-coloring of G, or a VDET coloring of G for short. The minimum number of colors required for a VDET colorings of G is denoted by X^evt（G）, and it is called the VDET chromatic number of G. In this article, we will discuss vertex-distinguishing E-total colorings of the graphs mC3 and mC4.
文摘讨论了m个阶为4的圈的点不交的并图m C 4的点可区别I-全染色和点可区别VI-全染色.通过构造以色集合和空集为元素的矩阵,利用色集合事先分配法及具体的染色给出了m C 4的最优点可区别I-全染色以及最优点可区别VI-全染色,进而确定了图m C 4的点可区别I-全色数及点可区别VI-全色数.结论表明点可区别I-全染色猜想和点可区别VI-全染色猜想对图m C 4成立.
