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**1 - 3**of**3**### Internal Partitions of Regular Graphs

, 2014

"... Published online in Wiley Online Library (wileyonlinelibrary.com). DOI 10.1002/jgt.21909 Abstract: An internal partition of an n-vertex graph G = (V,E) is a par-tition of V such that every vertex has at least as many neighbors in its own part as in the other part. It has been conjectured that every ..."

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Published online in Wiley Online Library (wileyonlinelibrary.com). DOI 10.1002/jgt.21909 Abstract: An internal partition of an n-vertex graph G = (V,E) is a par-tition of V such that every vertex has at least as many neighbors in its own part as in the other part. It has been conjectured that every d-regular graph with n> N(d) vertices has an internal partition. Here we prove this for d = 6. The case d = n − 4 is of particular interest and leads to inter-esting new open problems on cubic graphs. We also provide new lower bounds on N(d) and find new families of graphs with no internal partitions. Weighted versions of these problems are considered as well. C © 2015 Wiley

### 600 Olsen and Revsbæk On Alliance Partitions and Bisection Width for Planar Graphs

"... An alliance is a set of vertices (allies) such that any vertex in the alliance has at least as many allies (including the vertex itself) as non-allies in its neighborhood of the graph. The alliance is said to be strong if this holds even without including the vertex itself among the allies. Alliance ..."

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An alliance is a set of vertices (allies) such that any vertex in the alliance has at least as many allies (including the vertex itself) as non-allies in its neighborhood of the graph. The alliance is said to be strong if this holds even without including the vertex itself among the allies. Alliances of vertices in graphs [13] are used