Tight embeddings of partial quadrilateral packings

Completing partial packings of bipartite graphs

“…This cannot hold for odd-length paths, as can be seen by taking, for instance paths of length 3 and G = K 2,m , where m is large. Case k = 1 of the Conjecture is the special case of Proposition 9; it was first proved by Füredi and Lehel [4]. We think, we can also prove the Conjecture for k = 2.…”

“…Recall that a k-star is a copy of K 1,k . Case k = 2 was done by Füredi and Lehel [4]. We are using downdegree instead of updegree since this feels more natural to us.…”

“…Hilton and Lindner [7] were the first to prove a sub-linear bound on f (n; H) for a particular H. More precisely, they showed that a C 4 -packing can be completed by adding O(n 3/4 ) new vertices. Füredi and Lehel [4] found the right order of magnitude for f (n; C 4 ) by proving that f (n; C 4 ) = Θ( √ n). They conjectured that for any bipartite graph H the packing can be completed by adding o(n) new vertices.…”

“…A (by no means complete) list of references includes Hoffman, Küçükçifçi, Lindner, Roger, Stinson [8], [10], [11], [12], [13], [14], Jenkins [9], Bryant, Khodkar and El-Zanati [3]. See also Füredi and Lehel [4] for a survey of their results.…”

Small embeddings for partial 5‐cycle systems

“…For m = 4, it is known that . Füredi and Lehel recently showed that any partial 4‐cycle system of order u can be embedded in a 4‐cycle system of order . For even m ≥6, it is not clear what the lower bound on v should be; the smallest known integer v such that any partial m ‐cycle system of order u can be embedded in an m ‐cycle system of order v is um/ 2+ c ( m ), where c is a quadratic function of m , .…”