Counting Multigraph Pairs of λKn

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The purpose of this paper is to count multigraph pairs of λKn, as well as explore various results associated with multigraph pairs.

Let λKn be the complete λ-fold multigraph of order n. Have (G,H) be a λ-fold multigraph pair of order n such that:

1. There are no isolated vertices

2. G ≢ H

3. E(G) U E(H) = λKn

In this paper, we found the number of multigraph pairs of orders 2 and 3 of index λ. Additionally, we were able to find partial results for order 4 of index λ, as well as some relations of multigraphs based on structures of subgraphs of K4, involving counting and coloring.



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