Abstract
Let G be a graph with vertex set V(G) and edge set E(G) and let g and f be two integer-valuated functions defined on V(G) such that g(x) ≤f(x) for all x∈V(G). Then a (g, f)-factor of G is a spanning subgraph H of G such that g(x) ≤d H (x) ≤f(x) for all x∈V(G). A (g, f)-factorization of G is a partition of E(G) into edge-disjoint (g, f)-factors. Let \(\cal F\) = {F 1, F 2, ..., F m } be a factorization of G and H be a subgraph of G with mr edges. If F i , 1 ≤i≤m, has exactly r edges in common with H, then \(\cal F\) is said to be r-orthogonal to H. In this paper it is proved that every (mg + kr, mf−kr)-graph, where m, k and r are positive integers with k < m and g≥r, contains a subgraph R such that R has a (g, f)-factorization which is r-orthogonal to a given subgraph H with kr edges.
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This research is supported by the National Natural Science Foundation of China (19831080) and RSDP of China
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Li, G.J., Liu, G.Z. A Generalization of Orthogonal Factorizations in Graphs. Acta Math Sinica 17, 669–678 (2001). https://doi.org/10.1007/s101140100129
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DOI: https://doi.org/10.1007/s101140100129