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 𝝁𝑳𝒐 A decomposition of 𝜇𝐿𝑜 into cycles is a set of cycles that are subgraphs of 𝜇𝐿𝑜 whose edge sets partition the edge set of 𝜇𝐿𝑜. Obvious requirements:

• the sum of the cycle lengths be equal to the number of edges in 𝜇𝐿𝑜;
• the degree of each vertex of 𝜇𝐿𝑜 to be even.

Example: 2𝐿4 In the case, where the degree is not even, we decompose 𝜇𝐿𝑜 into cycles and a perfect matching.

For 𝑜 ≥ 3, 𝜇 ≥ 1, to partition the edge set of 𝜇𝐿𝑜 into t cycles of lengths 𝑛1, 𝑛2, … , 𝑛𝑢 , or into t cycles of lengths 𝑛1, 𝑛2, … , 𝑛𝑢 and a perfect matching, we require that: A list of integers 𝑛1, 𝑛2, … , 𝑛𝑢 that satisfy the above conditions for particular values of 𝜇 and 𝑜 is said to be (𝜇, 𝑜)–admissible. For shorthand, if 𝑁 = 𝑛1, 𝑛2, … , 𝑛𝑢, the notation (M)*-decomposition of 𝜇𝐿𝑜 will be used to denote both a decomposition of 𝜇𝐿𝑜into t cycles of lengths 𝑛1, 𝑛2, … , 𝑛𝑢 and a decomposition of 𝜇𝐿𝑜 into t cycles of lengths 𝑛1, 𝑛2, … , 𝑛𝑢 and a perfect matching. Cycle decompositions of the complete multigraph 𝝁𝑳𝒐 : necessary conditions

• 𝑛1 + 𝑛2 + ⋯ + 𝑛𝑢 = 𝜇 𝑜

2 when 𝜇(𝑜 − 1) is even;

• 𝑛1 + 𝑛2 + ⋯ + 𝑛𝑢 = 𝜇 𝑜

2

−

𝑜 2 when 𝜇 𝑜 − 1

is odd;

• 𝑛𝑗>2 𝑛𝑗 ≥ 𝑜

𝑜−1 2

when 𝜇 is odd.

• 2 ≤ 𝑛1, 𝑛2, … , 𝑛𝑢 ≤ 𝑜;

Theorem: Let 𝜇, n and m be integers with 𝑜, 𝑛 ≥ 3 and 𝜇 ≥ 1. There exists a decomposition of 𝜇𝐿𝑜 into cycles of length m if and only if 𝑛 ≤ 𝑜; 𝜇 𝑜 − 1 𝑗𝑡 𝑓𝑤𝑓𝑜; 𝑏𝑜𝑒 𝑛 𝑒𝑗𝑤𝑗𝑒𝑓𝑡 𝜇 𝑜 2 . There exists a decomposition of 𝜇𝐿𝑜 into cycles of length m and a perfect matching if and only if 𝑛 ≤ 𝑜; 𝜇 𝑜 − 1 𝑗𝑡 𝑝𝑒𝑒; 𝑏𝑜𝑒 𝑛 𝑒𝑗𝑤𝑗𝑒𝑓𝑡 𝜇 𝑜 2 − 𝑜/2. A very brief history of the problem of decomposing 𝜇𝐿𝑜 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 𝝁𝑳𝒐 : constant length cycles

Long Cycle Theorem: Let n ≥ 3 and 𝜇 be positive integers and let 𝑁 = 𝑛1, 𝑛2, … , 𝑛𝑢 be a (𝜇, n)-admissible list

• f integers. If 𝑛𝑗 ≥ ⌊

𝑜+3 2 ⌋ for 𝑗 = 1, 2, … , 𝑢, then there exists an (M)*-decomposition of 𝜇𝐿𝑜.

Short Cycle Theorem: Let n ≥ 3 and 𝜇 be positive integers and let 𝑁 = 𝑛1, 𝑛2, … , 𝑛𝑢 be a non-decreasing (𝜇, n)-admissible list of integers such that either 𝑛𝑢 = 𝑛𝑢−1 ≤ ⌊

𝑜+1 2 ⌋ or 𝑛𝑢 = 𝑛𝑢−1 + 1 ≤ ⌊ 𝑜+2 2 ⌋. Then there

exists an (M)*-decomposition of 𝜇𝐿𝑜. 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 3𝐿10. For example, there exists a (3,3,3,3,3,3,4,4,4,5,5)*-decomposition of 𝐿10 and a (3,3,3,3,3,3,3,4,4,5,6)*-decomposition of 𝐿10. Cycle decompositions of the complete multigraph 𝝁𝑳𝒐 : Two theorems for mixed length cycles

Theorem (Balister) Let 𝜇 and n be positive integers with 𝑜 ≥ 3 and 𝜇 𝑜 − 1 even. There exists a decomposition of 𝜇𝐿𝑜 into t closed trails of lengths 𝑛1, 𝑛2, … , 𝑛𝑢 if and only if

• 2 ≤ 𝑛1, 𝑛2, … , 𝑛𝑢;
• 𝑛1 + 𝑛2 + ⋯ + 𝑛𝑢 = 𝜇 𝑜

2 ; and

• 𝑛𝑗>2 𝑛𝑗 ≥

𝑜 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 𝜇𝐿𝑜 into t closed trails of lengths 𝑛1, 𝑛2, … , 𝑛𝑢 and a perfect matching if and only if

• 2 ≤ 𝑛1, 𝑛2, … , 𝑛𝑢;
• 𝑛1 + 𝑛2 + ⋯ + 𝑛𝑢 = 𝜇 𝑜

2

−

𝑜 2 ; and

• 𝑛𝑗>2 𝑛𝑗 ≥

𝑜 2

−

𝑜 2 .

Plan: For 𝑁 = 𝑛1, 𝑛2, … , 𝑛𝑢, to get an (M)*-decomposition of 𝜇𝐿𝑜, we start with a decomposition of 𝜇𝐿𝑜 into closed trails of lengths 𝑛1, 𝑛2, … , 𝑛𝑢 (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 𝑛1, 𝑛2, … , 𝑛𝑢 (and maybe a perfect matching). Cycle decompositions of the complete multigraph 𝝁𝑳𝒐 : Start with decompositions into closed trails

Let 𝑛1, 𝑛2, … , 𝑛𝑢 be a non-decreasing (𝜇, 𝑜)-admissible list, in which 𝑛𝑢 = 𝑛𝑢−1 ≤ ⌊

𝑜+1 2 ⌋ or 𝑛𝑢 = 𝑛𝑢−1 + 1 ≤ ⌊ 𝑜+2 2 ⌋.

To find an 𝑛1, 𝑛2, … , 𝑛𝑢 ∗-decomposition of 𝜇𝐿𝑜 : Start with a decomposition of 𝜇𝐿𝑜 into closed trails of lengths 𝑛1, 𝑛2, … , 𝑛𝑢. If these all happen to be cycles, you are done. If the biggest closed trails (lengths 𝑛𝑢−1 and 𝑛𝑢) 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 𝑛𝑢−1 and 𝑛𝑢. (This can be done since 𝑛𝑢−1 and 𝑛𝑢 differ by at most one.) Use edge switches to join the almost vertex-disjoint cycles into a chain of cycles. Then for each remaining closed trail (say of length 𝑛𝑗) that is not a cycle: Add the cycle of length 𝑛𝑢 to the cycle chain as the leave of a packing and use edge switches to obtain a cycle of length 𝑛𝑢 and a cycle of length 𝑛𝑗. Cycle decompositions of the complete multigraph 𝝁𝑳𝒐 : Outline of proof for short cycle theorem 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 𝑏1, 𝑏2, … , 𝑏𝑡, where 𝑏1 + 𝑏2 + ⋯ + 𝑏𝑡 = 𝑛𝑗.

Thank you for your attention!