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Range of diameters of complementary factors of almost complete tripartite graphs
Abstract
A complete tripartite graph without one edge, K̃m1,m2,m3, is called almost complete tripartite graph. A graph that K̃m1,m2,m3 that can be decomposed into two isomorphic factors with a given diameter d is called d-halvable. We prove that K̃m1,m2,m3 is d-halvable for a finite d only if d ≤ 5.
... In order to shorten the paper, we do not present the proofs of the lemmas here. The proofs can be found in [8]. We proceed step by step to show that if˜e =xy is the missing edge, then the actual distance between x and y must be greater than 5. ...
A complete tripartite graph without one edge, (K) over tilde (m1,m2,m3), is called almost complete tripartite graph. A graph (K) over tilde (m1),(m2),(m3) that can be decomposed into two isomorphic factors with a given diameter d is called d-halvable. We prove that (K) over tilde (m1,m2,m3) is d-halvable for a finite d only if d less than or equal to 5 and completely determine all triples 2m ' (1) + 1, 2m ' (2) +1, 2m ' (3) for which there exist d-halvable almost complete tripartite graphs for diameters 3,4 and 5, respectively.
We completely determine the spectrum of the complete bipartite and tripartite graphs that are decomposable into two isomorphic factors with a finite diameter.
A complete bipartite graph without one edge, <(K)over tilde (n,m)>, is called almost complete bipartite graph. A graph <(K)over tilde (2n+1,2m+1)> that can be decomposed into two isomorphic factors with st given diameter d is called d-isodecomposable. We prove that <(K)over tilde (2n+1,2m+1)> is d-isodecomposable only if d=3, 4, 5, 6 or infinity and completely determine all d-isodecomposable almost complete bipartite graphs for each diameter. For d=infinity we, moreover, present all classes of possible disconnected factors.