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1.
C. Balbuena 《Discrete Mathematics》2006,306(16):1817-1829
Let G be a graph of order n and size e. A vertex-magic total labeling is an assignment of the integers 1,2,…,n+e to the vertices and the edges of G, so that at each vertex, the vertex label and the labels on the edges incident at that vertex, add to a fixed constant, called the magic number of G. Such a labeling is a-vertex consecutive magic if the set of the labels of the vertices is {a+1,a+2,…,a+n}, and is b-edge consecutive magic if the set of labels of the edges is {b+1,b+2,…,b+e}. In this paper we prove that if an a-vertex consecutive magic graph has isolated vertices then the order and the size satisfy (n-1)2+n2=(2e+1)2. Moreover, we show that every tree with even order is not a-vertex consecutive magic and, if a tree of odd order is a-vertex consecutive then a=n-1. Furthermore, we show that every a-vertex consecutive magic graph has minimum degree at least two if a=0, or both and 2a?e, and the minimum degree is at least three if both and 2a?e. Finally, we state analogous results for b-edge consecutive magic graphs. 相似文献
2.
For all integers m3 and all natural numbers a1,a2,…,am−1, let n=R(a1,a2,…,am−1) represent the least integer such that for every 2-coloring of the set {1,2,…,n} there exists a monochromatic solution to
a1x1+a2x2++am−1xm−1=xm.