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On endo-homology of complexes of graphs. (English) Zbl 0955.05104
Summary: Let \(L\) be a subcomplex of a complex \(K\). If the homomorphism from inclusion \(i_*: H_q(L)\mapsto H_q(K)\) is an isomorphism for all \(q\geq 0\), then we say that \(L\) and \(K\) are endo-homologous. The clique complex of a graph \(G\), denoted by \(C(G)\), is an abstract complex whose simplices are the cliques of \(G\). The present paper is a generalization of A. V. Ivashchenko [Discrete Math. 126, No. 1-3, 159-170 (1994; Zbl 0798.05027)] along several directions. For a graph \(G\) and a given subgraph \(F\) of \(G\), some necessary and sufficient conditions for \(C(G)\) to be endo-homologous to \(C(F)\) are given. Similar theorems hold also for the independence complex \(I(G)\) of \(G\), where \(I(G)= C(G^{\text{c}})\), the clique complex of the complement of \(G\).
05C99 Graph theory
05E25 Group actions on posets, etc. (MSC2000)
Full Text: DOI
[1] Bondy, J.A.; Murty, U.S.R., Graph theory with applications, (1976), The Macmillan Press London · Zbl 1134.05001
[2] Giblin, P.J., Graphs, surfaces and homology, (1981), Chapman & Hall London · Zbl 0477.57001
[3] Ivashchenko, A.V., Contractible transformations do not change the homology groups of graphs, Discrete math., 126, 159-170, (1994) · Zbl 0798.05027
[4] Jiang, C.H., Introduction to topology, (1978), Shanghai Publishing Company of Science and Technology, (in Chinese)
[5] Lovasz, L., Kneser’s conjecture, chromatic number and homotopy, J. combin. theory ser. A, 25, 319-324, (1978) · Zbl 0418.05028
[6] Peng, Y., On the invariance of the homology groups of the neighbourhood complex of a simple graph, Ann. math., 11A, 6, 677-682, (1990), (in Chinese) · Zbl 0744.05019
[7] Xie, L.; Liu, J.; Liu, G., Graphs and algebraic topology, (1994), Shandong University Press, (in Chinese)
[8] L. Xie, On independence complexes of graphs. Graphs and Combinatorics’95, vol. 2, World Scientific Press, Singapore, to appear.
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