Rödl, Ruciński, and Szemerédi determined the minimum $(k-1)$-degree threshold for the existence of fractional perfect matchings in $k$-uniform hypergrahs, and Kühn, Osthus, and Townsend extended this result by asymptotically determining the $d$-degree threshold for the range $k-1>d\ge k/2$. In this note, we prove the following exact degree threshold: Let $k,d$ be positive integers with $k\ge 4$ and $k-1>d\geq k/2$, and let $n$ be any integer with $n\ge k^2$. Then any $n$-vertex $k$-uniform hypergraph with minimum $d$-degree $δ_d(H)>{n-d\choose k-d} -{n-d-(\lceil n/k\rceil-1)\choose
k-d}$ contains a fractional perfect matching. This lower bound on the minimum $d$-degree is best possible. We also determine optimal minimum $d$-degree conditions which guarantees the existence of fractional matchings of size $s$, where $0<s\le n/k$ (when $k/2\le d\le k-1$), or with $s$ large enough and $s\le n/k$ (when $2k/5<d<k/2$).
Extending the notion of (random) $k$-out graphs, we consider when the $k$-out hypergraph is likely to have a perfect fractional matching. In particular, we show that for each $r$ there is a $k=k(r)$ such that the $k$-out $r$-uniform hypergraph on $n$ vertices has a perfect fractional matching with high probability (i.e., with probability tending to $1$ as $n\to \infty$) and prove an analogous result for $r$-uniform $r$-partite hypergraphs. This is based on a new notion of hypergraph expansion and the observation that sufficiently expansive hypergraphs admit perfect fractional matchings. As a f…
Perfect state transfer and fractional revival can be used to move information between pairs of vertices in a quantum network. While perfect state transfer has received a lot of attention, fractional revival is newer and less studied. One problem is to determine the differences between perfect state transfer and fractional revival. If perfect state transfer occurs between two vertices in a graph, the vertices must be cospectral. Further if there is perfect state transfer between vertices $a$ and $b$ in a graph, there cannot be perfect state transfer from $a$ to any other vertex. No examples of …
The restricted hypercube-like graphs, variants of the hypercube, were proposed as desired interconnection networks of parallel systems. The matching preclusion number of a graph is the minimum number of edges whose deletion results in the graph with neither perfect matchings nor almost perfect matchings. The fractional perfect matching preclusion and fractional strong perfect matching preclusion are generalizations of the concept matching preclusion. In this paper, we obtain fractional matching preclusion number and fractional strong matching preclusion numbers of restricted hypercube-like gra…
It has been asked whether there are trees other than $P_2$ and $P_3$ which can admit perfect state transfers. In this note we show that the answer is negative.
Helly's theorem is a classical result concerning the intersection patterns of convex sets in $\mathbb{R}^d$. Two important generalizations are the colorful version and the fractional version. Recently, Bárány et al. combined the two, obtaining a colorful fractional Helly theorem. In this paper, we give an improved version of their result.
In this paper we prove Ahlswede- Khachatrian conjecture~\cite{1} up to finite number of cases, which can be checked using modern computers. From this conjecture follows conjecture from~\cite{2} and Manickam-Miklós-Singhi conjecture. This paper is combined from papers~\cite{09},~\cite{08}.
The strong matching preclusion number of a graph, introduced by Park and Ihm in 2011, is the minimum number of vertices and edges whose deletion results in a graph that has neither perfect matchings nor almost perfect matchings. As a generalization, the fractional strong matching preclusion number of a graph is the minimum number of edges and vertices whose deletion leaves the resulting graph without a fractional perfect matching. In this paper, we obtain the fractional strong matching preclusion number for Cartesian product graphs. As an application, the fractional strong matching preclusion …
We determine the \emph{exact} minimum $\ell$-degree threshold for perfect matchings in $k$-uniform hypergraphs when the corresponding threshold for perfect fractional matchings is significantly less than $\frac{1}{2} \binom{n}{k- \ell}$. This extends our previous results that determine the minimum $\ell$-degree thresholds for perfect matchings in $k$-uniform hypergraphs for all $\ell\ge k/2$ and provides two new (exact) thresholds: $(k,\ell)=(5,2)$ and $(7,3)$.
A fractional colouring of a graph $G$ is a function that assigns a non-negative real value to all possible colour-classes of $G$ containing any vertex of $G$, such that the sum of these values is at least one for each vertex. The fractional chromatic number is the minimum sum of the values assigned by a fractional colouring over all possible such colourings of $G$. Introduced by Bosica and Tardif, fractional covers are an extension of fractional colourings whereby the real-valued function acts on all possible subgraphs of $G$ belonging to a given class of graphs. The fractional chromatic numbe…
The fractional perfect b-matching polytope of an undirected graph G is the polytope of all assignments of nonnegative real numbers to the edges of G such that the sum of the numbers over all edges incident to any vertex v is a prescribed nonnegative number b_v. General theorems which provide conditions for nonemptiness, give a formula for the dimension, and characterize the vertices, edges and face lattices of such polytopes are obtained. Many of these results are expressed in terms of certain spanning subgraphs of G which are associated with subsets or elements of the polytope. For example, i…
In this paper, we generalize fractional $q$-integrals by the method of $q$-difference equation. In addition, we deduce fractional Askey--Wilson integral, reversal type fractional Askey--Wilson integral and Ramanujan type fractional Askey--Wilson integral.
Matching preclusion is a measure of robustness in the event of edge failure in interconnection networks. As a generalization of matching preclusion, the fractional matching preclusion number (FMP number for short) of a graph is the minimum number of edges whose deletion results in a graph that has no fractional perfect matchings, and the fractional strong matching preclusion number (FSMP number for short) of a graph is the minimum number of edges and/or vertices whose deletion leaves a resulting graph with no fractional perfect matchings. A graph $G$ is said to be $f$-fault Hamiltonian if ther…
We conjecture that every oriented graph $G$ on $n$ vertices with $δ^+ (G) , δ^- (G) \geq 5n/12$ contains the square of a Hamilton cycle. We also give a conjectural bound on the minimum semidegree which ensures a perfect packing of transitive triangles in an oriented graph. A link between Ramsey numbers and perfect packings of transitive tournaments is also considered.
We are given a bipartite graph that contains at least one perfect matching and where each edge is colored from a set $Q=\{c_1,c_2,\ldots,c_q}\$. Let $Q_i=\set{e\in E(G):c(e)=c_i}$, where $c(e)$ denotes the color of $e$. The perfect matching color profile $mcp(G)$ is defined to be the set of vectors $(m_1,m_2,\ldots,m_q)\in [n]^q$ such that there exists a perfect matching $M$ such that $|M\cap Q_i|=m_i$. We give bounds on the matching color profile for a randomly colored random bipartite graph.
The \emph{fractional matching preclusion number} of a graph $G$, denoted by $fmp(G)$, is the minimum number of edges whose deletion results in a graph that has no fractional perfect matchings. In this paper, we first give some sharp upper and lower bounds of fractional matching preclusion number. Next, graphs with large and small fractional matching preclusion number are characterized, respectively. In the end, we investigate some extremal problems on fractional matching preclusion number.
The boxicity of a graph $G$ is the minimum non-negative integer $k$ such that $G$ can be isomorphic to the intersection graph of a family of boxes in Euclidean $k$-space, where a box in Euclidean $k$-space is the Cartesian product of $k$ closed intervals on the real line. In this note, we define the fractional boxicity of a graph as the optimum value of the linear relaxation of a covering problem with respect to boxicity, which gives a lower bound for its boxicity. We show that the fractional boxicity of a graph is at least the lower bounds for boxicity given by Adiga et al. in 2014. We also p…
We reprove that all the matchings constructed during Edmonds' weighted perfect matching algorithm are optimal among those of the same cardinality (provided that certain mild restrictions are obeyed on the choices the algorithm makes). We conclude that in order to solve a weighted matching problem it is not needed to solve a weighted perfect matching problem in an auxiliary graph of doubled size. This result was known before, e.g., posed as an exercise in see Lawler's book from 1976, but is not present in several modern books on combinatorial optimization.
In this paper we study conditions which guarantee the existence of perfect matchings and perfect fractional matchings in uniform hypergraphs. We reduce this problem to an old conjecture by Erdős on estimating the maximum number of edges in a hypergraph when the (fractional) matching number is given, which we are able to solve in some special cases using probabilistic techniques. Based on these results, we obtain some general theorems on the minimum $d$-degree ensuring the existence of perfect (fractional) matchings. In particular, we asymptotically determine the minimum vertex degree which gua…
