Switching techniques for edge decompositions of graphs Daniel - PowerPoint PPT Presentation

Switching in triangle packings

Switching in triangle packings

Adding a triangle to a packing

Adding a triangle to a packing

Adding a triangle to a packing Say we wish to add another triangle to a partial embedding. The leave of our packing has no triangles, but does contain a “lasso” on the new vertices.

Adding a triangle to a packing Say we wish to add another triangle to a partial embedding. The leave of our packing has no triangles, but does contain a “lasso” on the new vertices.

Adding a triangle to a packing Say we wish to add another triangle to a partial embedding. The leave of our packing has no triangles, but does contain a “lasso” on the new vertices.

Adding a triangle to a packing Say we wish to add another triangle to a partial embedding. The leave of our packing has no triangles, but does contain a “lasso” on the new vertices.

Adding a triangle to a packing Say we wish to add another triangle to a partial embedding. The leave of our packing has no triangles, but does contain a “lasso” on the new vertices.

Adding a triangle to a packing Say we wish to add another triangle to a partial embedding. The leave of our packing has no triangles, but does contain a “lasso” on the new vertices.

Adding a triangle to a packing Say we wish to add another triangle to a partial embedding. The leave of our packing has no triangles, but does contain a “lasso” on the new vertices.

Theorem (Bryant, H. 2009) Every PSTS ( u ) has an embedding of order v for each admissible v � 2 u + 1.

More switching-assisted results on embeddings Bryant, Buchanan (2007): Every partial totally symmetric quasigroup of order u has an embedding of order v for each even v � 2 u + 4. Bryant, Martin (2012): For u � 28, every PTS ( u , λ ) has an embedding triple of order v for each admissible v � 2 u + 1. Martin, McCourt (2012): Any partial 5-cycle system of order u � 255 has an embedding of order at most 1 4 ( 9 u + 146 ) . H. (2014): “Half” of the possible embeddings of order less than 2 u + 1 for PSTS ( u ) s with ∆( L ) � 1 1 4 ( u − 9 ) and | E ( L ) | < 32 ( u − 5 )( u − 11 ) + 2 exist. 50 u 2 + o ( u ) triples has an embedding 1 H. (2014): Any PSTS ( u ) with at most for each admissible order v � 1 5 ( 8 u + 17 ) .

Part 2: Cycle decompositions

Cycle decompositions of complete graphs

Cycle decompositions of complete graphs K n � � � m 1 , . . . , m t “There is a decomposition of K n into cycles of lengths m 1 , . . . , m t .”

Cycle decompositions of complete graphs K n � � � m 1 , . . . , m t “There is a decomposition of K n into cycles of lengths m 1 , . . . , m t .”

Cycle decompositions of complete graphs K n � � � m 1 , . . . , m t “There is a decomposition of K n into cycles of lengths m 1 , . . . , m t .” K 7 � 7 , 6 , 4 , 4

If K n � m 1 , . . . , m t then (1) n is odd; (2) n � m 1 , . . . , m t � 3; and � n � (3) m 1 + · · · + m t = . 2

If K n � m 1 , . . . , m t then (1) n is odd; (2) n � m 1 , . . . , m t � 3; and � n � (3) m 1 + · · · + m t = . 2 Alspach’s cycle decomposition problem (1981) Prove (1), (2) and (3) are sufficient for K n � m 1 , . . . , m t .

History (highlights)

History (highlights) When does K n � m , . . . , m ?

History (highlights) When does K n � m , . . . , m ? ◮ Kirkman and Walecki solved special cases in the 1800s. ◮ Results from Kotzig, Rosa, Huang in the 1960s. ◮ Reductions of the problem from Bermond, Huang, Sotteau and from Hoffman, Lindner, Rodger in the 1980s and 90s. ◮ Solved by Alspach, Gavlas, ˇ Sajna in 2001–2002.

History (highlights) When does K n � m , . . . , m ? ◮ Kirkman and Walecki solved special cases in the 1800s. ◮ Results from Kotzig, Rosa, Huang in the 1960s. ◮ Reductions of the problem from Bermond, Huang, Sotteau and from Hoffman, Lindner, Rodger in the 1980s and 90s. ◮ Solved by Alspach, Gavlas, ˇ Sajna in 2001–2002. When does K n � m 1 , . . . , m t ?

History (highlights) When does K n � m , . . . , m ? ◮ Kirkman and Walecki solved special cases in the 1800s. ◮ Results from Kotzig, Rosa, Huang in the 1960s. ◮ Reductions of the problem from Bermond, Huang, Sotteau and from Hoffman, Lindner, Rodger in the 1980s and 90s. ◮ Solved by Alspach, Gavlas, ˇ Sajna in 2001–2002. When does K n � m 1 , . . . , m t ? ◮ Work on limited sets of cycle lengths from Adams, Bryant, Heinrich, Hor´ ak, Khodkar, Maehaut, Rosa in the 1980s, 90s and 00s. ◮ A more general result from Balister in 2001. ◮ A reduction from Bryant, H. in 2009–2010. ◮ Solved by Bryant, H., Pettersson in 2014.

Switching in cycle packings

Switching in cycle packings