Maximum packings of the complete graph with uniform length cycles

Abstract

Abstract:In this paper we find the maximum number of pairwise edgedisjoint m-cycles which exist in a complete graph with n vertices, for all values of n and m with 3 ≤ m ≤ n. ᭧

Decomposing Various Graphs into Short Even-Length Cycles

“…, m − 2, m + 2} such that |E(G 1 )| ≡ e (mod m). It follows from Lemma 2.5 of [15] (noting that v − u + 1 ≥ 7) that there is an m-cycle packing P 1 of G 1 whose leave has an e-cycle as its only non-trivial component if e = 0 and whose leave is empty if e = 0. To complete the proof we will find an m-cycle packing of G 2 with a leave L 2 such that the union of L 2 with the leave of P 1 (after a relabelling of the vertices in P 1 ) has a decomposition into…”

“…, m − 2, m + 2} such that |E(G 1 )| ≡ e (mod m). It follows from Lemma 2.5 of [15] (noting that v − u + 1 ≥ 7) that there is an m-cycle packing P 1 of G 1 whose leave has an e-cycle as its only non-trivial component if e = 0 and whose leave is empty if e = 0. To complete the proof we will find an m-cycle packing of G 2 with a leave L 2 such that the union of L 2 with the leave of P 1 (after a relabelling of the vertices in P 1 ) has a decomposition into We claim that we can relabel the vertices in P 1 in such a way that its leave has a decomposition into a 2-path P 1 from y to z and an (e−2)-path Q 1 from y to z such that We claim that we can relabel the vertices in P 1 in such a way that its leave has a decomposition into a 4-path P 1 from y to z and an (m − 2)-path Q 1 from y to z such that V (P 1 ) ∩ V (P 2 ) = {y, z} and V (Q 1 ) ∩ V (Q 2 ) = {y, z}.…”

Group divisible ‐packings with any minimum leave

“…Maximum packings of have attracted the most attention (see [2][3][4][6][7][8]14,17] for example). Maximum packings of have attracted the most attention (see [2][3][4][6][7][8]14,17] for example).…”

“…Maximum packings of graphs have been and continue to be popular topics of research. Maximum packings of have attracted the most attention (see [2][3][4][6][7][8]14,17] for example). However, very little has been done on maximum packings of ( ) .…”

Cycle packings of the complete multigraph

“…In the case of the complete graph K v (with λ = 1), it had previously been found exactly when there exist decompositions into cycles of specified lengths [6]. Furthermore, Horsley [10] found conditions for the existence of packings of the complete graph with uniform length cycles. These results built on earlier results for cycle decompositions and packings of the complete graph [1,2,9,11] (see [7] for a survey).…”