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Title: Transversal Ck -factors in subgraphs of the balanced blow-up of Ck
Abstract

For a subgraph$G$of the blow-up of a graph$F$, we let$\delta ^*(G)$be the smallest minimum degree over all of the bipartite subgraphs of$G$induced by pairs of parts that correspond to edges of$F$. Johansson proved that if$G$is a spanning subgraph of the blow-up of$C_3$with parts of size$n$and$\delta ^*(G) \ge \frac{2}{3}n + \sqrt{n}$, then$G$contains$n$vertex disjoint triangles, and presented the following conjecture of Häggkvist. If$G$is a spanning subgraph of the blow-up of$C_k$with parts of size$n$and$\delta ^*(G) \ge \left(1 + \frac 1k\right)\frac n2 + 1$, then$G$contains$n$vertex disjoint copies of$C_k$such that each$C_k$intersects each of the$k$parts exactly once. A similar conjecture was also made by Fischer and the case$k=3$was proved for large$n$by Magyar and Martin.

In this paper, we prove the conjecture of Häggkvist asymptotically. We also pose a conjecture which generalises this result by allowing the minimum degree conditions in each bipartite subgraph induced by pairs of parts of$G$to vary. We support this new conjecture by proving the triangle case. This result generalises Johannson’s result asymptotically.

 
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Award ID(s):
1800761
NSF-PAR ID:
10475779
Author(s) / Creator(s):
;
Publisher / Repository:
Combinatorics, Probability, & Computing
Date Published:
Journal Name:
Combinatorics, Probability and Computing
Volume:
31
Issue:
6
ISSN:
0963-5483
Page Range / eLocation ID:
1031 to 1047
Subject(s) / Keyword(s):
extremal graph theory cycles
Format(s):
Medium: X
Sponsoring Org:
National Science Foundation
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