Dense, irregular, yet always graphic $3$-uniform hypergraph degree sequences (2312.00555v1)
Abstract: A $3$-uniform hypergraph is a generalization of simple graphs where each hyperedge is a subset of vertices of size $3$. The degree of a vertex in a hypergraph is the number of hyperedges incident with it. The degree sequence of a hypergraph is the sequence of the degrees of its vertices. The degree sequence problem for $3$-uniform hypergraphs is to decide if a $3$-uniform hypergraph exists with a prescribed degree sequence. Such a hypergraph is called a realization. Recently, Deza \emph{et al.} proved that the degree sequence problem for $3$-uniform hypergraphs is NP-complete. Some special cases are easy; however, polynomial algorithms have been known so far only for some very restricted degree sequences. The main result of our research is the following. If all degrees are between $\frac{2n2}{63}+O(n)$ and $\frac{5n2}{63}-O(n)$ in a degree sequence $D$, further, the number of vertices is at least $45$, and the degree sum can be divided by $3$, then $D$ has a $3$-uniform hypergraph realization. Our proof is constructive and in fact, it constructs a hypergraph realization in polynomial time for any degree sequence satisfying the properties mentioned above. To our knowledge, this is the first polynomial running time algorithm to construct a $3$-uniform hypergraph realization of a highly irregular and dense degree sequence.
- Half-regular factorizations of the complete bipartite graph. Discrete Applied Mathematics, 230, 21–33.
- Enumerating sparse uniform hypergraphs with given degree sequence and forbidden edges. European Journal of Combinatorics, 77, 68–77.
- Approximate sampling and counting of graphs with near-regular degree intervals. preprint, https://arxiv.org/abs/2110.09068, Accepted at STACS 2023.
- Characterization and reconstruction of hypergraphic pattern sequences. IWCIA, pages 301–316.
- Bollobás, B. (1980). A probabilistic proof of an asymptotic formula for the number of regular graphs. European Journal of Combinatorics, 1, 311–314.
- Optimization over degree sequences. SIAM Journal on Discrete Mathematics, 32, 2067–2079.
- Hypergraphic degree sequences are hard. https://arxiv.org/pdf/1901.02272.pdf.
- 3-colored grids from horizontal and vertical projections is np-hard. European Symposium on Algorithms, pages 776–787.
- The mixing time of the switch markov chains: a unified approach. European Journal of Combinatorics, 99, 103421.
- New classes of degree sequences with fast mixing swap markov chain sampling. Combinatorics, Probability and Computing, 27(2), 186–207.
- Graphs with vertices of prescribed degrees (in hungarian). Matematikai Lapok, 11, 264–274.
- On the degree sequences of uniform hypergraphs. In: Gonzalez-Diaz, R., Jimenez, M.J., Medrano, B. (eds.) Discrete Geometry for Computer Imagery. DGCI 2013. Lecture Notes in Computer Science, 7749, 300–310.
- New sufficient conditions on the degree sequences of uniform hypergraphs. Theor. Comput. Sci., 868, 97–111.
- Gale, D. (1957). A theorem on flows in networks. Pacific J. Math., 2, 1073–1082.
- Packing tree degree sequences. Graphs and Combinatorics, 36, 779–801.
- Hakimi, S. (1962). On the realizability of a set of integers as degrees of the vertices of a simple graph. J. SIAM Appl. Math., 10, 779–801.
- Havel, V. (1955). A remark on the existence of finite graphs. (czech). Časopis Pěst. Mat., 80, 477–480.
- Kundu, S. (1974). Disjoint representation of tree realizable sequences. SIAM Journal on Applied Mathematics, 26(1), 103–107.
- Kundu, S. (1975). Disjoint representation of three tree realizable sequences. SIAM J. of Appl. Math., 28, 290–302.
- Edge disjoint caterpillar realizations. Discrete Applied Mathematics, 289, 327–344.
- On the reconstruction of 3-uniform hypergraphs from degree sequences of span-two. Journal of Mathematical Imaging and Vision, 64, 693–704.
- Ryser, H. (1957). Combinatorial properties of matrices of zeros and ones. Canad. J. Math., 9, 371–377.