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1. A Unified Proof of Conjectures on Cycle Lengths in Graphs

   https://doi.org/10.1093/imrn/rnaa324  

   Gao, Jun; Huo, Qingyi; Liu, Chun-Hung; Ma, Jie (January 2021, International Mathematics Research Notices)

   null (Ed.)

   Abstract In this paper, we prove a tight minimum degree condition in general graphs for the existence of paths between two given endpoints whose lengths form a long arithmetic progression with common difference one or two. This allows us to obtain a number of exact and optimal results on cycle lengths in graphs of given minimum degree, connectivity or chromatic number. More precisely, we prove the following statements by a unified approach: 1. Every graph $$G$$ with minimum degree at least $k+1$ contains cycles of all even lengths modulo $$k$$; in addition, if $$G$$ is $$2$$-connected and non-bipartite, then it contains cycles of all lengths modulo $$k$$. 2. For all $$k\geq 3$$, every $$k$$-connected graph contains a cycle of length zero modulo $$k$$. 3. Every $$3$$-connected non-bipartite graph with minimum degree at least $k+1$ contains $$k$$ cycles of consecutive lengths. 4. Every graph with chromatic number at least $k+2$ contains $$k$$ cycles of consecutive lengths. The 1st statement is a conjecture of Thomassen, the 2nd is a conjecture of Dean, the 3rd is a tight answer to a question of Bondy and Vince, and the 4th is a conjecture of Sudakov and Verstraëte. All of the above results are best possible. 

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2. Small Even Covers, Locally Decodable Codes and Restricted Subgraphs of Edge-Colored Kikuchi Graphs

   https://doi.org/10.1093/imrn/rnaf045  

   Hsieh, Jun-Ting; Kothari, Pravesh K; Mohanty, Sidhanth; Correia, David Munhá; Sudakov, Benny (March 2025, International Mathematics Research Notices)

   Abstract Given a $$k$$-uniform hypergraph $$H$$ on $$n$$ vertices, an even cover in $$H$$ is a collection of hyperedges that touch each vertex an even number of times. Even covers are a generalization of cycles in graphs and are equivalent to linearly dependent subsets of a system of linear equations modulo $$2$$. As a result, they arise naturally in the context of well-studied questions in coding theory and refuting unsatisfiable $$k$$-SAT formulas. Analogous to the irregular Moore bound of Alon, Hoory, and Linial [3], Feige conjectured [8] an extremal trade-off between the number of hyperedges and the length of the smallest even cover in a $$k$$-uniform hypergraph. This conjecture was recently settled up to a multiplicative logarithmic factor in the number of hyperedges [12, 13]. These works introduce the new technique that relates hypergraph even covers to cycles in the associated Kikuchi graphs. Their analysis of these Kikuchi graphs, especially for odd $$k$$, is rather involved and relies on matrix concentration inequalities. In this work, we give a simple and purely combinatorial argument that recovers the best-known bound for Feige’s conjecture for even $$k$$. We also introduce a novel variant of a Kikuchi graph which together with this argument improves the logarithmic factor in the best-known bounds for odd $$k$$. As an application of our ideas, we also give a purely combinatorial proof of the improved lower bounds [4] on 3-query binary linear locally decodable codes. 

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3. Codegree Conditions for Tiling Complete k-Partite k-Graphs and Loose Cycles

   https://doi.org/10.1017/S096354831900021X  

   Gao, Wei; Han, Jie; Zhao, Yi (November 2019, Combinatorics, Probability and Computing)

   Abstract Given two k -graphs ( k -uniform hypergraphs) F and H , a perfect F -tiling (or F -factor) in H is a set of vertex-disjoint copies of F that together cover the vertex set of H . For all complete k -partite k -graphs K , Mycroft proved a minimum codegree condition that guarantees a K -factor in an n -vertex k -graph, which is tight up to an error term o ( n ). In this paper we improve the error term in Mycroft’s result to a sublinear term that relates to the Turán number of K when the differences of the sizes of the vertex classes of K are co-prime. Furthermore, we find a construction which shows that our improved codegree condition is asymptotically tight in infinitely many cases, thus disproving a conjecture of Mycroft. Finally, we determine exact minimum codegree conditions for tiling K (k) (1, … , 1, 2) and tiling loose cycles, thus generalizing the results of Czygrinow, DeBiasio and Nagle, and of Czygrinow, respectively. 

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4. Schubert defects in Lagrangian Grassmannians

   https://doi.org/10.1007/JHEP06(2025)148  

   Gu, Wei; Mihalcea, Leonardo; Sharpe, Eric; Xu, Weihong; Zhang, Hao; Zou, Hao (June 2025, Journal of High Energy Physics)

   A<sc>bstract</sc> In this paper, we propose a construction of GLSM defects corresponding to Schubert cycles in Lagrangian Grassmannians, following recent work of Closset-Khlaif on Schubert cycles in ordinary Grassmannians. In the case of Lagrangian Grassmannians, there are superpotential terms in both the bulk GLSM as well as on the defect itself, enforcing isotropy constraints. We check our construction by comparing the locus on which the GLSM defect is supported to mathematical descriptions, checking dimensions, and perhaps most importantly, comparing defect indices to known and expected polynomial invariants of the Schubert cycles in quantum cohomology and quantum K theory. 

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5. On the Tanner Cycle Distribution of QC-LDPC Codes from Polynomial Parity-Check Matrices

   https://doi.org/10.1109/isit54713.2023.10207005  

   Gómez-Fonseca, Anthony; Smarandache, Roxana; Mitchell, David G. (June 2023, 023 IEEE International Symposium on Information Theory (ISIT))

   In this paper, we present an efficient strategy to enumerate the number of k-cycles, g ≤ k < +2g, in the Tanner graph of a quasi-cyclic low-density parity-check (QC-LDPC) code with girth g using its polynomial parity-check matrix H. This strategy works for both (n c , n v )-regular and irregular QC-LDPC codes. In this approach, we note that the mth power of the polynomial adjacency matrix can be used to describe walks of length m in the protograph and can therefore be sufficiently described by the matrices Bm(H)≜(HH⊤)⌊m/2⌋H(mmod2), where m ≥ 0. For example, in the case of QC-LDPC codes based on the 3 × n v fully-connected protograph, the complexity of determining the number of k-cycles, Nk, for k = 4, 6 and 8, is O(n2vlog(N)), O(n2vlog(nv)log(N)) and O(n4vlog4(nv)log(N)), respectively. The complexity, depending logarithmically on the lifting factor N, gives our approach, to the best of our knowledge, a significant advantage over previous works on the cycle distribution of QC-LDPC codes. 

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