Packing and Covering Triangles in Bilaterally-Complete Tripartite Graphs

We use Menger's Theorem and K\"onig's Line Colouring Theorem to show that in any tripartite graph with two complete (bipartite) sides the maximum number of pairwise edge-disjoint triangles equals the minimum number of edges that meet all triangles. This generalizes the corresponding result for complete tripartite graphs given by Lakshmanan, et al.Comment: 8 pages, 4 figure

Sufficient Conditions for Tuza's Conjecture on Packing and Covering Triangles

Given a simple graph $G=(V,E)$, a subset of $E$ is called a triangle cover if it intersects each triangle of $G$. Let $\nu_t(G)$ and $\tau_t(G)$ denote the maximum number of pairwise edge-disjoint triangles in $G$ and the minimum cardinality of a triangle cover of $G$, respectively. Tuza conjectured in 1981 that $\tau_t(G)/\nu_t(G)\le2$ holds for every graph $G$. In this paper, using a hypergraph approach, we design polynomial-time combinatorial algorithms for finding small triangle covers. These algorithms imply new sufficient conditions for Tuza's conjecture on covering and packing triangles. More precisely, suppose that the set $\mathscr T_G$ of triangles covers all edges in $G$. We show that a triangle cover of $G$ with cardinality at most $2\nu_t(G)$ can be found in polynomial time if one of the following conditions is satisfied: (i) $\nu_t(G)/|\mathscr T_G|\ge\frac13$, (ii) $\nu_t(G)/|E|\ge\frac14$, (iii) $|E|/|\mathscr T_G|\ge2$. Keywords: Triangle cover, Triangle packing, Linear 3-uniform hypergraphs, Combinatorial algorithm

Induced cycles in triangle graphs

The triangle graph of a graph $G$, denoted by ${\cal T}(G)$, is the graph whose vertices represent the triangles ($K_3$ subgraphs) of $G$, and two vertices of ${\cal T}(G)$ are adjacent if and only if the corresponding triangles share an edge. In this paper, we characterize graphs whose triangle graph is a cycle and then extend the result to obtain a characterization of $C_n$-free triangle graphs. As a consequence, we give a forbidden subgraph characterization of graphs $G$ for which ${\cal T}(G)$ is a tree, a chordal graph, or a perfect graph. For the class of graphs whose triangle graph is perfect, we verify a conjecture of the third author concerning packing and covering of triangles.Comment: 27 page

Towards Ryser\u27s Conjecture: Bounds on the Cardinality of Partitioned Intersecting Hypergraphs

This work is motivated by the open conjecture concerning the size of a minimum vertex cover in a partitioned hypergraph. In an r-uniform r-partite hypergraph, the size of the minimum vertex cover C is conjectured to be related to the size of its maximum matching M by the relation (|C|\u3c= (r-1)|M|). In fact it is not known whether this conjecture holds when |M| = 1. We consider r-partite hypergraphs with maximal matching size |M| = 1, and pose a novel algorithmic approach to finding a vertex cover of size (r - 1) in this case. We define a reactive hypergraph to be a back-and-forth algorithm for a hypergraph which chooses new edges in response to a choice of vertex cover, and prove that this algorithm terminates for all hypergraphs of orders r = 3 and 4. We introduce the idea of optimizing the size of the reactive hypergraph and find that the reactive hypergraph terminates for r = 5...20. We then consider the case where the intersection of any two edges is exactly 1. We prove bounds on the size of this 1-intersecting hypergraph and relate the 1-intersecting hypergraph maximization problem to mutually orthogonal Latin squares. We propose a generative algorithm for 1-intersecting hypergraphs of maximal size for prime powers r-1 = pd under the constraint pd+1 is also a prime power of the same form, and therefore pose a new generating algorithm for MOLS based upon intersecting hypergraphs. We prove this algorithm generates a valid set of mutually orthogonal Latin squares and prove the construction guarantees certain symmetric properties. We conclude that a conjecture by Lovasz, that the inequality in Ryser\u27s Conjecture cannot be improved when (r-1) is a prime power, is correct for the 1-intersecting hypergraph of prime power orders

Matchings and Covers in Hypergraphs

In this thesis, we study three variations of matching and covering problems in hypergraphs. The first is motivated by an old conjecture of Ryser which says that if \mcH is an $r$-uniform, $r$-partite hypergraph which does not have a matching of size at least $\nu +1$, then \mcH has a vertex cover of size at most $(r-1)\nu$. In particular, we examine the extremal hypergraphs for the $r=3$ case of Ryser's conjecture. In 2014, Haxell, Narins, and Szab{\'{o}} characterized these $3$-uniform, tripartite hypergraphs. Their work relies heavily on topological arguments and seems difficult to generalize. We reprove their characterization and significantly reduce the topological dependencies. Our proof starts by using topology to show that every $3$-uniform, tripartite hypergraph has two matchings which interact with each other in a very restricted way. However, the remainder of the proof uses only elementary methods to show how the extremal hypergraphs are built around these two matchings. Our second motivational pillar is Tuza's conjecture from 1984. For graphs $G$ and $H$, let $\nu_{H}(G)$ denote the size of a maximum collection of pairwise edge-disjoint copies of $H$ in $G$ and let $\tau_{H}(G)$ denote the minimum size of a set of edges which meets every copy of $H$ in $G$. The conjecture is relevant to the case where $H=K_{3}$ and says that $\tau_{\triangledown}(G) \leq 2\nu_{\triangledown}(G)$ for every graph $G$. In 1998, Haxell and Kohayakawa proved that if $G$ is a tripartite graph, then $\tau_{\triangledown}(G) \leq 1.956\nu_{\triangledown}(G)$. We use similar techniques plus a topological result to show that $\tau_{\triangledown}(G) \leq 1.87\nu_{\triangledown}(G)$ for all tripartite graphs $G$. We also examine a special subclass of tripartite graphs and use a simple network flow argument to prove that $\tau_{\triangledown}(H) = \nu_{\triangledown}(H)$ for all such graphs $H$. We then look at the problem of packing and covering edge-disjoint $K_{4}$'s. Yuster proved that if a graph $G$ does not have a fractional packing of $K_{4}$'s of size bigger than $\nu_{\boxtimes}^{*}(G)$, then $\tau_{\boxtimes}(G) \leq 4\nu_{\boxtimes}^{*}(G)$. We give a complementary result to Yuster's for $K_{4}$'s: We show that every graph $G$ has a fractional cover of $K_{4}$'s of size at most $\frac{9}{2}\nu_{\boxtimes}(G)$. We also provide upper bounds on $\tau_{\boxtimes}$ for several classes of graphs. Our final topic is a discussion of fractional stable matchings. Tan proved that every graph has a $\frac{1}{2}$-integral stable matching. We consider hypergraphs. There is a natural notion of fractional stable matching for hypergraphs, and we may ask whether an analogous result exists for this setting. We show this is not the case: Using a construction of Chung, F{\"{u}}redi, Garey, and Graham, we prove that, for all n \in \mbN, there is a $3$-uniform hypergraph with preferences with a fractional stable matching that is unique and has denominators of size at least $n$

Packing And Covering Triangles In Tripartite Graphs

. It is shown that if G is a tripartite graph such that the maximum size of a set of pairwise edge-disjoint triangles is (G), then there is a set C of edges of G of size at most (2 ")(G) such that E(T ) \ C 6= ; for every triangle T of G, where " > 0:044. This improves the previous bound of (7=3)(G) due to Tuza [6]. x0. Introduction In this note we are concerned with a packing and covering problem for triangles in graphs. Let a graph G be given. We say that a family F of triangles in G is independent if the elements of F are pairwise edge-disjoint. Moreover, we say that a set C E(G) of edges of G is a transversal of the set of triangles of G, or simply a transversal for G, if every triangle of G contains at least one element of C. We denote by (G) the maximum cardinality of an independent family of triangles in G, and by (G) the minimum cardinality of a transversal for G. It is clear that for every graph G we have (G) (G) 3(G). The graphs G = K 4 and K 5 show tha..