Fractional refinements of integral theorems by Benjamin Richard Moore A thesis presented to the University of Waterloo in fulfillment of the thesis requirement for the degree of Doctor of Philosophy in Combinatorics and Optimization Waterloo, Ontario, Canada, 2021 © Benjamin Richard Moore 2021 Examining Committee Membership The following served on the Examining Committee for this thesis. The decision of the Examining Committee is by majority vote. External Examiner: Gary MacGillivray Professor, Dept. of Mathematics and Statistics, University of Victoria Supervisor: Luke Postle Professor, Dept. of Combinatorics and Optimization, University of Waterloo Internal Member: Sophie Spirkl Professor, Dept. of Combinatorics and Optimization, University of Waterloo Internal Member: Bertrand Guenin Professor, Dept. of Combinatorics and Optimization, University of Waterloo Internal-External Member: Ross Willard Professor, Dept. of Pure Mathematics, University of Waterloo ii Author's Declaration This thesis consists of material all of which I authored or co-authored: see Statement of Contributions included in the thesis. This is a true copy of the thesis, including any required final revisions, as accepted by my examiners. I understand that my thesis may be made electronically available to the public. iii Statement of Contributions This thesis contains joint work with Logan Grout, Evelyne Smith-Roberge, Richard Brewster, and Douglas B. West. In particular, Chapter 2 is based on work with Logan Grout, Chapter 3 and 4 is based on work with Evelyne Smith Roberge, Chapter 6 is based on work with Richard Brewster, and Chapter 7 is based on work with Douglas B. West. iv Abstract The focus of this thesis is to take theorems which deal with \integral" objects in graph theory and consider fractional refinements of them to gain additional structure. A classic theorem of Hakimi says that for an integer k, a graph has maximum average degree at most 2k if and only if the graph decomposes into k pseudoforests. To find a fractional refinement of this theorem, one simply needs to consider the instances where the maximum average degree is fractional. We prove that for any positive integers k and d, if G has maximum average degree at 2d most 2k + k+d+1 , then G decomposes into k + 1 pseudoforests, where one of pseudoforests has every connected component containing at most d edges, and further this pseudoforest is acyclic. The maximum average degree bound is best possible for every choice of k and d. Similar to Hakimi's Theorem, a classical theorem of Nash-Williams says that a graph has fractional arborcity at most k if and only if G decomposes into k forests. The Nine Dragon Tree Theorem, proven by Jiang and Yang, provides a fractional refinement of Nash- Williams Theorem. It says, for any positive integers k and d, if a graph G has fractional d arboricity at most k + k+d+1 , then G decomposes into k +1 forests, where one of the forests has maximum degree d. We prove a strengthening of the Nine Dragon Tree Theorem in certain cases. Let k = 1 d and d 2 f3; 4g. Every graph with fractional arboricity at most 1 + d+2 decomposes into two forests T and F where F has maximum degree d, every component of F contains at most one vertex of degree d, and if d = 4, then every component of F contains at most 8 edges e = xy such that both deg(x) ≥ 3 and deg(y) ≥ 3. 3 In fact, when k = 1 and d = 3, we prove that every graph with fractional arboricity 1+ 5 decomposes into two forests T;F such that F has maximum degree 3, every component of F has at most one vertex of degree 3, further if a component of F has a vertex of degree 3 then it has at most 14 edges, and otherwise a component of F has at most 13 edges. Shifting focus to problems which partition the vertex set, circular colouring provides a way to fractionally refine colouring problems. A classic theorem of Tuza says that ev- ery graph with no cycles of length 1 mod k is k-colourable. Generalizing this to circular colouring, we get the following: Let k and d be relatively prime, with k > 2d, and let s be the element of Zk such that sd ≡ 1 mod k. Let xy be an edge in a graph G. If G − xy is (k; d)-circular-colorable and G is not, then xy lies in at least one cycle in G of length congruent to is mod k for some i in f1; : : : ; dg. If this does not occur with i 2 f1; : : : ; d − 1g, then xy lies in at least two cycles of length 1 mod k and G − xy contains a cycle of length 0 mod k. This theorem is best possible with regards to the number of congruence classes when k = 2d + 1. v A classic theorem of Gr¨otzsch says that triangle free planar graphs are 3-colourable. There are many generalizations of this result, however fitting the theme of fractional re- finements, Jaeger conjectured that every planar graph of girth 4k admits a homomorphism to C2k+1. While we make no progress on this conjecture directly, one way to approach the conjecture is to prove critical graphs have large average degree. On this front, we prove: Every 4-critical graph which does not have a (7; 2)-colouring and is not K4 or W5 17v(G) 5v(G)+2 satisfies e(G) ≥ 10 , and every triangle free 4-critical graph satisfies e(G) ≥ 3 . In the case of the second theorem, a result of Davies shows there exists infinitely many 5v(G)+4 triangle free 4-critical graphs satisfying e(G) = 3 , and hence the second theorem is close to being tight. It also generalizes results of Thomas and Walls, and also Thomassen, that girth 5 graphs embeddable on the torus, projective plane, or Klein bottle are 3- colourable. Lastly, a theorem of Cereceda, Johnson, and van den Heuvel, says that given a 2- connected bipartite planar graph G with no separating four-cycles and a 3-colouring f, then one can obtain all 3-colourings from f by changing one vertices' colour at a time if and only if G has at most one face of size 6. p We give the natural generalization of this to circular colourings when q < 4. vi Acknowledgements There are many people to thank during my graduate studies. The single best thing to come out of my graduate studies is meeting other graduate students. It is hard to imagine a better graduate student community than what is here in Waterloo. In particular, I would like to thank Kelvin Chan for organizing many parties, baking much bread, planning many camping trips, and watching alot of matroids. I also would like to thank Charupriya Sharma for many very fun times. She was instrumental in making Waterloo fun. Others are also very deserving of individual shout outs, but to do them justice I would have to make the acknowledgements longer than my thesis, and so simply say thanks Alan Wong, Florian Hoersch, Lukas Nabergall, Iain Crump, Sabrina Lato, Ronen Wdowinski, Tim Miller, Martin Pei, Tina Chen, Rose McCarty, Nick Olsen Harris, Matthew Kroeker, Sina Baghal, Amena Assem, Brett Nassaraden, Arnott Kidner, Soffi´a Arnad´ottir,Jason´ Qu, Tao Jiang, James Davies and Kazuhiro Nomoto for being great (and anyone I missed by accident as well!) For this thesis, I would also like to thank two (ex)-graduate students of C and O in particular, Evelyne Smith Roberge and Logan Grout, who contributed greatly to content of this thesis, and without whom it is not clear the thesis would have been completed (at least without many errors). Some profs and postdocs are also deserving of thanks. I would like to thank Carl Feghali for many discussions and opportunities. I would like to thank my undergrad supervisors Rick Brewster and Sean Mcguinness for helping me many times. I would like to thank Naomi Nishimura for the many meetings. I would especially like to thank Joseph Cheriyan for giving me the opportunity to study at Waterloo, introducing me to Logan, having lots of good advice, and also having the good idea of thinking about pseudoforests. I would also like to thank my readers Sophie Spirkl, Bertrand Guenin, Gary MacGilli- vary and Ross Willard for valuable comments on this thesis. I would also like to thank Xuding Zhu, Daqing Yang and Doug West for allowing me to visit Zheijang Normal University, and their many good ideas which helped in the making of this thesis. I would also like to thank Melissa Cambridge for much help throughout my graduate career. Finally I would like to thank NSERC for financial support throughout my graduate and undergraduate career. vii Dedication This thesis is dedicated to all my family members in Kamloops, especially my Mom and Dad. viii Table of Contents List of Figures xi 1 Introduction1 1.1 On the Nine Dragon Tree Theorem......................2 1.2 Colouring sparse graphs............................ 11 1.3 Cycles and paths between colourings..................... 19 1.4 Structure of the Thesis............................. 23 2 The Pseudoforest Strong Nine Dragon Tree Theorem 24 2.1 Introduction................................... 24 2.2 Picking the counterexample.......................... 26 2.3 Augmenting the decomposition........................ 31 2.4 Bounding the maximum average degree.................... 35 3 Strong Dragons are tough to slay 38 3.1 Introduction................................... 38 3.2 Picking the minimal counterexample..................... 38 3.3 Reducible Configurations............................ 44 3.4 The counting argument for non-bad components............... 54 3.5 The counting argument for bad components................. 59 3.6 Strengthening the d = 3; k = 1 case.....................