Cycle decompositions of complete multigraphs Barbara Maenhaut, The - PowerPoint PPT Presentation

Cycle decompositions of complete multigraphs Barbara Maenhaut, The University of Queensland Joint work with Darryn Bryant, Daniel Horsley and Ben Smith

Cycle decompositions of the complete multigraph 𝝁𝑳 𝒐 Example: 2𝐿 4 A decomposition of 𝜇𝐿 𝑜 into cycles is a set of cycles that are subgraphs of 𝜇𝐿 𝑜 whose edge sets partition the edge set of 𝜇𝐿 𝑜 . Obvious requirements: the sum of the cycle lengths be equal to the number of edges in 𝜇𝐿 𝑜 ; • the degree of each vertex of 𝜇𝐿 𝑜 to be even. • In the case, where the degree is not even, we decompose 𝜇𝐿 𝑜 into cycles and a perfect matching.

Cycle decompositions of the complete multigraph 𝝁𝑳 𝒐 : necessary conditions For 𝑜 ≥ 3, 𝜇 ≥ 1, t o partition the edge set of 𝜇𝐿 𝑜 into t cycles of lengths 𝑛 1 , 𝑛 2 , … , 𝑛 𝑢 , or into t cycles of lengths 𝑛 1 , 𝑛 2 , … , 𝑛 𝑢 and a perfect matching, we require that: 2 ≤ 𝑛 1 , 𝑛 2 , … , 𝑛 𝑢 ≤ 𝑜; • 𝑛 1 + 𝑛 2 + ⋯ + 𝑛 𝑢 = 𝜇 𝑜 2 when 𝜇(𝑜 − 1) is even; • 𝑜 𝑛 1 + 𝑛 2 + ⋯ + 𝑛 𝑢 = 𝜇 𝑜 • − 2 when 𝜇 𝑜 − 1 is odd; 2 𝑜−1 𝑛 𝑗 >2 𝑛 𝑗 ≥ 𝑜 when 𝜇 is odd. • 2 A list of integers 𝑛 1 , 𝑛 2 , … , 𝑛 𝑢 that satisfy the above conditions for particular values of 𝜇 and 𝑜 i s said to be (𝜇, 𝑜) – admissible. For shorthand, if 𝑁 = 𝑛 1 , 𝑛 2 , … , 𝑛 𝑢 , the notation ( M )*-decomposition of 𝜇𝐿 𝑜 will be used to denote both a decomposition of 𝜇𝐿 𝑜 into t cycles of lengths 𝑛 1 , 𝑛 2 , … , 𝑛 𝑢 and a decomposition of 𝜇𝐿 𝑜 into t cycles of lengths 𝑛 1 , 𝑛 2 , … , 𝑛 𝑢 and a perfect matching.

Cycle decompositions of the complete multigraph 𝝁𝑳 𝒐 : constant length cycles Theorem: Let 𝜇, n and m be integers with 𝑜, 𝑛 ≥ 3 and 𝜇 ≥ 1 . There exists a decomposition of 𝜇𝐿 𝑜 into cycles of length m if and only if 𝑜 𝑛 ≤ 𝑜; 𝜇 𝑜 − 1 𝑗𝑡 𝑓𝑤𝑓𝑜; 𝑏𝑜𝑒 𝑛 𝑒𝑗𝑤𝑗𝑒𝑓𝑡 𝜇 2 . There exists a decomposition of 𝜇𝐿 𝑜 into cycles of length m and a perfect matching if and only if 𝑜 𝑛 ≤ 𝑜; 𝜇 𝑜 − 1 𝑗𝑡 𝑝𝑒𝑒; 𝑏𝑜𝑒 𝑛 𝑒𝑗𝑤𝑗𝑒𝑓𝑡 𝜇 2 − 𝑜/2. A very brief history of the problem of decomposing 𝜇𝐿 𝑜 into m -cycles (or into m -cycles and a perfect matching): The case 𝜇 = 1: Many specific cases solved over many years, but finally solved by Alspach, Gavlas and Šajna (2001, 2002). • The case 𝜇 = 2: Solved by Alspach, Gavlas, Šajna and Verall by considering decompositions into directed cycles (2003). • • The cases 𝜇 ≥ 3: 𝑛 = 3 (Hanani, 1961), 4 ≤ 𝑛 ≤ 6 (Huang and Rosa, 1973, 1975), 8 ≤ 𝑛 ≤ 16, m even (Bermond, Huang and Sotteau, 1978), 3 ≤ 𝑛 ≤ 7, 𝑛 odd (Bermond and Sotteau, 1977), m an odd prime (Smith, 2010) 𝜇 a multiple of m (Smith, 2010), n odd and 𝜇𝑜 a multiple of m (Smith, 2010).

Cycle decompositions of the complete multigraph 𝝁𝑳 𝒐 : Two theorems for mixed length cycles Long Cycle Theorem: Let n ≥ 3 and 𝜇 be positive integers and let 𝑁 = 𝑛 1 , 𝑛 2 , … , 𝑛 𝑢 be a ( 𝜇 , n )-admissible list 𝑜+3 of integers. If 𝑛 𝑗 ≥ ⌊ 2 ⌋ for 𝑗 = 1, 2, … , 𝑢 , then there exists an ( M )*-decomposition of 𝜇𝐿 𝑜 . For example, there exists a (6,6,6,6,6,7,7,7,7,8,9,9,9,9,9,9,9,10)*-decomposition of 3𝐿 10 . Short Cycle Theorem: Let n ≥ 3 and 𝜇 be positive integers and let 𝑁 = 𝑛 1 , 𝑛 2 , … , 𝑛 𝑢 be a non-decreasing 𝑜+1 𝑜+2 ( 𝜇 , n )-admissible list of integers such that either 𝑛 𝑢 = 𝑛 𝑢−1 ≤ ⌊ 2 ⌋ or 𝑛 𝑢 = 𝑛 𝑢−1 + 1 ≤ ⌊ 2 ⌋ . Then there exists an ( M )*-decomposition of 𝜇𝐿 𝑜 . For example, there exists a (3,3,3,3,3,3,4,4,4,5,5)*-decomposition of 𝐿 10 and a (3,3,3,3,3,3,3,4,4,5,6)*-decomposition of 𝐿 10 .

Cycle decompositions of the complete multigraph 𝝁𝑳 𝒐 : Start with decompositions into closed trails Theorem (Balister) Let 𝜇 and n be positive integers with 𝑜 ≥ 3 and 𝜇 𝑜 − 1 even. There exists a decomposition of 𝜇𝐿 𝑜 into t closed trails of lengths 𝑛 1 , 𝑛 2 , … , 𝑛 𝑢 if and only if 2 ≤ 𝑛 1 , 𝑛 2 , … , 𝑛 𝑢 ; • 𝑛 1 + 𝑛 2 + ⋯ + 𝑛 𝑢 = 𝜇 𝑜 2 ; and • 𝑜 𝑛 𝑗 >2 𝑛 𝑗 ≥ 2 when 𝜇 is odd. • Theorem Let 𝜇 and n be positive integers with 𝑜 ≥ 4 and 𝜇 𝑜 − 1 odd. There exists a decomposition of 𝜇𝐿 𝑜 into t closed trails of lengths 𝑛 1 , 𝑛 2 , … , 𝑛 𝑢 and a perfect matching if and only if • 2 ≤ 𝑛 1 , 𝑛 2 , … , 𝑛 𝑢 ; 𝑜 𝑛 1 + 𝑛 2 + ⋯ + 𝑛 𝑢 = 𝜇 𝑜 − 2 ; and • 2 𝑜 𝑜 𝑛 𝑗 >2 𝑛 𝑗 ≥ − • 2 . 2 Plan: For 𝑁 = 𝑛 1 , 𝑛 2 , … , 𝑛 𝑢 , t o get an ( M )*-decomposition of 𝜇𝐿 𝑜 , we start with a decomposition of 𝜇𝐿 𝑜 into closed trails of lengths 𝑛 1 , 𝑛 2 , … , 𝑛 𝑢 (and maybe a perfect matching). Split the closed trails into cycles and then manipulate (modify and glue) these cycles in such a way as to get cycles of lengths 𝑛 1 , 𝑛 2 , … , 𝑛 𝑢 (and maybe a perfect matching).

Cycle decompositions of the complete multigraph 𝝁𝑳 𝒐 : Outline of proof for short cycle theorem 𝑜+1 𝑜+2 Let 𝑛 1 , 𝑛 2 , … , 𝑛 𝑢 be a non-decreasing (𝜇, 𝑜) -admissible list, in which 𝑛 𝑢 = 𝑛 𝑢−1 ≤ ⌊ 2 ⌋ or 𝑛 𝑢 = 𝑛 𝑢−1 + 1 ≤ ⌊ 2 ⌋ . To find an 𝑛 1 , 𝑛 2 , … , 𝑛 𝑢 ∗ -decomposition of 𝜇𝐿 𝑜 : Start with a decomposition of 𝜇𝐿 𝑜 into closed trails of lengths 𝑛 1 , 𝑛 2 , … , 𝑛 𝑢 . If these all happen to be cycles, you are done. If the biggest closed trails (lengths 𝑛 𝑢−1 and 𝑛 𝑢 ) are not cycles, delete those two closed trails from the decomposition (they become the leave of a packing) and use edge switches to make them into cycles of lengths 𝑛 𝑢−1 and 𝑛 𝑢 . (This can be done since 𝑛 𝑢−1 and 𝑛 𝑢 differ by at most one.) Then for each remaining closed trail (say of length 𝑛 𝑗 ) that is not a cycle: Delete the closed trail from the decomposition (it becomes the leave) and use edge switches to spread it out to a collection of almost vertex-disjoint cycles of lengths 𝑏 1 , 𝑏 2 , … , 𝑏 𝑡 , where 𝑏 1 + 𝑏 2 + ⋯ + 𝑏 𝑡 = 𝑛 𝑗 . Use edge switches to join the almost vertex-disjoint cycles into a chain of cycles. Add the cycle of length 𝑛 𝑢 to the cycle chain as the leave of a packing and use edge switches to obtain a cycle of length 𝑛 𝑢 and a cycle of length 𝑛 𝑗 .

Thank you for your attention!