Saturation numbers of books.

Chen, Guantao; Faudree, Ralph J.; Gould, Ronald J.

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Supermagic Graphs Having a Saturated Vertex

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A graph is called supermagic if it admits a labeling of the edges by pairwise different consecutive integers such that the sum of the labels of the edges incident with a vertex is independent of the particular vertex. In this paper we establish some conditions for graphs with a saturated vertex to be supermagic. Inter alia we show that complete multipartite graphs K1,n,n and K1,2,...,2 are supermagic.

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Henning, Michael A., Yeo, Anders (2007)

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On the Numbers of Cut-Vertices and End-Blocks in 4-Regular Graphs

Dingguo Wang, Erfang Shan (2014)

Discussiones Mathematicae Graph Theory

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A cut-vertex in a graph G is a vertex whose removal increases the number of connected components of G. An end-block of G is a block with a single cut-vertex. In this paper we establish upper bounds on the numbers of end-blocks and cut-vertices in a 4-regular graph G and claw-free 4-regular graphs. We characterize the extremal graphs achieving these bounds.