The rr-coloring and maximum stable set problem in hypergraphs with bounded matching number and edge size

Motivated by the analogous questions in graphs, we study the complexity of coloring and stable set problems in hypergraphs with forbidden substructures and bounded edge size. Letting $\nu(G)$ denote the maximum size of a matching in $H$, we obtain complete dichotomies for the complexity of the following problems parametrized by fixed $r, k, s \in \mathbb{N}$: $r$-Coloring in hypergraphs $G$ with edge size at most $k$ and $\nu(G) \leq s$; $r$-Precoloring Extension in $k$-uniform hypergraphs $G$ with $\nu(G) \leq s$; $r$-Precoloring Extension in hypergraphs $G$ with edge size at most $k$ and $\nu(G) \leq s$; Maximum Stable Set in $k$-uniform hypergraphs $G$ with $\nu(G) \leq s$; Maximum Weight Stable Set in $k$-uniform hypergraphs with $\nu(G) \leq s$; as well as partial results for $r$-Coloring in $k$-uniform hypergraphs $\nu(G) \leq s$. We then turn our attention to $2$-Coloring in 3-uniform hypergraphs with forbidden induced subhypergraphs, and give a polynomial-time algorithm when restricting the input to hypergraphs excluding a fixed one-edge hypergraph. Finally, we consider linear 3-uniform hypergraphs (in which every two edges share at most one vertex), and show that excluding an induced matching in $G$ implies that $\nu(G)$ is bounded by a constant; and that $3$-coloring linear $3$-uniform hypergraphs $G$ with $\nu(G) \leq 532$ is NP-hard

A family of extremal hypergraphs for Ryser's conjecture

Ryser's Conjecture states that for any $r$-partite $r$-uniform hypergraph, the vertex cover number is at most $r{-}1$ times the matching number. This conjecture is only known to be true for $r\leq 3$ in general and for $r\leq 5$ if the hypergraph is intersecting. There has also been considerable effort made for finding hypergraphs that are extremal for Ryser's Conjecture, i.e. $r$-partite hypergraphs whose cover number is $r-1$ times its matching number. Aside from a few sporadic examples, the set of uniformities $r$ for which Ryser's Conjecture is known to be tight is limited to those integers for which a projective plane of order $r-1$ exists. We produce a new infinite family of $r$-uniform hypergraphs extremal to Ryser's Conjecture, which exists whenever a projective plane of order $r-2$ exists. Our construction is flexible enough to produce a large number of non-isomorphic extremal hypergraphs. In particular, we define what we call the {\em Ryser poset} of extremal intersecting $r$-partite $r$-uniform hypergraphs and show that the number of maximal and minimal elements is exponential in $\sqrt{r}$. This provides further evidence for the difficulty of Ryser's Conjecture

Perfect Matchings, Tilings and Hamilton Cycles in Hypergraphs

This thesis contains problems in finding spanning subgraphs in graphs, such as, perfect matchings, tilings and Hamilton cycles. First, we consider the tiling problems in graphs, which are natural generalizations of the matching problems. We give new proofs of the multipartite Hajnal-Szemeredi Theorem for the tripartite and quadripartite cases. Second, we consider Hamilton cycles in hypergraphs. In particular, we determine the minimum codegree thresholds for Hamilton l-cycles in large k-uniform hypergraphs for l less than k/2. We also determine the minimum vertex degree threshold for loose Hamilton cycle in large 3-uniform hypergraphs. These results generalize the well-known theorem of Dirac for graphs. Third, we determine the minimum codegree threshold for near perfect matchings in large k-uniform hypergraphs, thereby confirming a conjecture of Rodl, Rucinski and Szemeredi. We also show that the decision problem on whether a k-uniform hypergraph with certain minimum codegree condition contains a perfect matching can be solved in polynomial time, which solves a problem of Karpinski, Rucinski and Szymanska completely. At last, we determine the minimum vertex degree threshold for perfect tilings of C_4^3 in large 3-uniform hypergraphs, where C_4^3 is the unique 3-uniform hypergraph on four vertices with two edges

On Positive Matching Decomposition Conjectures of Hypergraphs

In this paper, we prove the conjectures of Gharakhloo and Welker (2023) that the positive matching decomposition number (pmd) of a $3$-uniform hypergraph is bounded from above by a polynomial of degree $2$ in terms of the number of vertices. Moreover, we derive a lower bound for pmd specifically for complete $3$-uniform hypergraphs. Additionally, we obtain an upper bound for pmd of $r$-uniform hypergraphs. For a $r$-uniform hypergraphs $H=(V,E)$ such that $\lvert e_i\cap e_j\rvert \leq 1$ for all $e_i,e_j \in E$, we give a characterization of positive matching in terms of strong alternate closed walks. For specific classes of a hypergraph, we classify the radical and complete intersection Lov\'{a}sz$-$Saks$-$Schrijver ideals.Comment: Subsection 2.1 is adde

Coloring Algorithms for Graphs and Hypergraphs with Forbidden Substructures

This thesis mainly focus on complexity results of the generalized version of the $r$-Coloring Problem, the $r$-Pre-Coloring Extension Problem and the List $r$-Coloring Problem restricted to hypergraphs and ordered graphs with forbidden substructures. In the context of forbidding non-induced substructure in hypergraphs, we obtain complete complexity dichotomies of the $r$-Coloring Problem and the $r$-Pre-Coloring Extension Problem in hypergraphs with bounded edge size and bounded matching number, as well as the $r$-Pre-Coloring Extension Problem in hypergraphs with uniform edge size and bounded matching number. We also get partial complexity result of the $r$-Coloring Problem in hypergraphs with uniform edge size and bounded matching number. Additionally, we study the Maximum Stable Set Problem and the Maximum Weight Stable Set Problem in hypergraphs. We obtain complexity dichotomies of these problems in hypergraphs with uniform edge size and bounded matching number. We then give a polynomial-time algorithm of the 2-Coloring Problem restricted to the class of 3-uniform hypergraphs excluding a fixed one-edge induced subhypergraph. We also consider linear hypergraphs and show that 3-Coloring in linear 3-uniform hypergraphs with either bounded matching size or bounded induced matching size is NP-hard if the bound is a large enough constant. This thesis also contains a near-dichotomy of complexity results for ordered graphs. We prove that the List-3-Coloring Problem in ordered graphs with a forbidden induced ordered subgraph is polynomial-time solvable if the ordered subgraph contains only one edge, or it is isomorphic to some fixed ordered 3-vertex path plus isolated vertices. On the other hand, it is NP-hard if the ordered subgraph contains at least three edges, or contains a vertex of degree two and does not satisfy the polynomial-time case mentioned before, or contains two non-adjacent edges with a specific ordering. The complexity result when forbidding a few ordered subgraphs with exactly two edges is still unknown

Exact minimum degree thresholds for perfect matchings in uniform hypergraphs II

Given positive integers k\geq 3 and r where k/2 \leq r \leq k-1, we give a minimum r-degree condition that ensures a perfect matching in a k-uniform hypergraph. This condition is best possible and improves on work of Pikhurko who gave an asymptotically exact result. Our approach makes use of the absorbing method, and builds on work in 'Exact minimum degree thresholds for perfect matchings in uniform hypergraphs', where we proved the result for k divisible by 4.Comment: 20 pages, 1 figur