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Regular bipartite graph

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Consider the sets A0 := {0, 1, 4}, B0 := {0, 2, 8}. Consider the sets Ai := A0 + i := {i, i + 1, i + 4} ,and
Bi := B0 + i := {i, i + 2, i + 8}, for i = 1, 2, . . . , 12. All addition here is performed modulo 13. Consider the bipartite graph G whose vertices are the sets A := {Ai : 0 <= i <= 12} and B := {Bi : 0 <= i <= 12} (so that the graph has a total of 26 vertices) and vertices corresponding to two different sets Ai , Bj are adjacent in the graph G if and only if the intersection of Ai and Bj is the empty set.

1. Check that this bipartite graph is regular.
2. The regularity of the graph implies that the edges of G can be
partitioned into edge disjoint perfect matchings. Give one such
partition.

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Solution Summary

The content of each vertex of G, as well as the set of edges of G, are determined. That information is used to show that G is regular. Also, a partition of the edges of G into edge disjoint perfect matchings is provided.

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Georgia Martin, PhD
Education
  • AB, Hood College
  • PhD, The Catholic University of America
  • PhD, The University of Maryland at College Park
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