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Four-Cycled Graphs with Topological Applications
Türker Bıyıkoglu1 and Yusuf Civan2
1Department of Mathematics, Is¸ik University, S¸ ile 34980, Istanbul, Turkey
turker.biyikoglu@isikun.edu.tr
2Department of Mathematics, Suleyman Demirel University, Isparta 32260, Turkey
ycivan@fef.sdu.edu.tr
Annals of Combinatorics 16 (1) pp.37-56 March, 2012
AMS Subject Classification: 57Q10, 05C38; 05C75
Abstract:
We call a simple graph G a 4-cycled graph if either it has no edges or every edge of it is contained in an induced 4-cycle of G. Our interest on 4-cycled graphs is motivated by the fact that their clique complexes play an important role in the simple-homotopy theory of simplicial complexes. We prove that the minimal simple models within the category of flag simplicial complexes are exactly the clique complexes of some 4-cycled graphs. We further provide structural properties of 4-cycled graphs and describe constructions yielding such graphs. We characterize 4-cycled cographs, and 4-cycled graphs arising from finite chessboards. We introduce
a family of inductively constructed graphs, the external extensions, related to an arbitrary graph, and determine the homotopy type of the independence complexes of external extensions of some graphs.
Keywords: clique and independence complexes, cycled graph, cograph, chessboard graph, simple-homotopy, s-homotopy

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