Fractional Perfect Matching Research Articles

Conditional Fractional Matching Preclusion Number of Graphs

The conditional fractional matching preclusion number (CFMP number for short) [Formula: see text] of a graph [Formula: see text] is the minimum number of edges whose deletion results in a graph without isolated vertices and without fractional perfect matchings. In this paper, we study the CFMP number of complete graphs, complete bipartite graphs and twisted cubes. Also, we give Nordhaus–Gaddum-type results for the CFMP numbers of general graphs.

Distance spectral radius and fractional matching in t-connected graphs

A fractional matching of a graph G is a function f assigning each edge a number in [ 0 , 1 ] so that ∑ e ∈ Γ ( v ) f ( e ) ≤ 1 for each v ∈ V ( G ) , where Γ ( v ) is the set of edges incident to v. The fractional matching number is the maximum of ∑ e ∈ Γ ( v ) f ( e ) over all fractional matchings. Motivated by progress in the study of relations between eigenvalues and matchings of graphs, in this paper, we characterize graphs with the minimum distance spectral radius among all t-connected graphs with n vertices and fractional matching number at most n − k 2 for 1 ≤ k ≤ n − 2 . Our characterization generalizes a result of Li, Miao, and Zhang [On the size, spectral radius, distance spectral radius and fractional matchings in graphs. Bull Aust Math Soc. 2023;187–199.], giving a distance spectral condition for the existence of a fractional perfect matching in a connected graph.

Fractional matching preclusion numbers of Cartesian product graphs

The Cartesian product of two simple graphs G and H is the graph G□H whose vertex set is V(G)×V(H) and whose edge set is the set of all pairs (u1,v1)(u2,v2) such that either u1u2∈E(G) and v1=v2, or v1v2∈E(H) and u1=u2. The fractional matching preclusion number of a graph G, denoted by fmp(G), is the minimum number of edges whose deletion results in a graph with no fractional perfect matching. In this paper, we determine fmp(G□H) when H is a cycle or a path of even order; Moreover, given any integers a,b with a≥1 and 0≤b≤a+1, we construct a graph G such that δ(G)=a and fmp(G□H)=b when H is a path of odd order.

Fractional matching, factors and spectral radius in graphs involving minimum degree

A fractional matching of a graph G is a function f:E(G)→[0,1] such that for any v∈V(G), ∑e∈EG(v)f(e)≤1, where EG(v)={e∈E(G):eis incident withvinG}. The fractional matching number of G is μf(G)=max{∑e∈E(G)f(e):f is a fractional matching of G}. Let k∈(0,n) is an integer. In this paper, we prove a tight lower bound of the spectral radius to guarantee μf(G)>n−k2 in a graph with minimum degree δ, which implies the result on the fractional perfect matching due to Fan et al. (2022) [6].For a set {A,B,C,…} of graphs, an {A,B,C,…}-factor of a graph G is defined to be a spanning subgraph of G each component of which is isomorphic to one of {A,B,C,…}. We present a tight sufficient condition in terms of the spectral radius for the existence of a {K2,{Ck}}-factor in a graph with minimum degree δ, where k≥3 is an integer. Moreover, we also provide a tight spectral radius condition for the existence of a {K1,1,K1,2,…,K1,k}-factor with k≥2 in a graph with minimum degree δ, which generalizes the result of Miao et al. (2023) [10].

Spectral Radius and Fractional Perfect Matchings in Graphs

Spectral Radius and Fractional Perfect Matchings in Graphs

A Factor Matching of Optimal Tail Between Poisson Processes

Consider two independent Poisson point processes of unit intensity in the Euclidean space of dimension d at least 3. We construct a perfect matching between the two point sets that is a factor (i.e., a measurable function of the point configurations that commutes with translations), and with the property that the distance between two matched configuration points has a tail distribution that decays as fast as possible in magnitude, namely, as bexp (-cr^d) with suitable constants b,c>0. This settles the most difficult version of such matching problems: bicolored (versus unicolored) and deterministic (versus randomized). Our proof relies on two earlier results: an allocation (“land-division”) rule of similar tail for a Poisson point process by Markó and the author, and a recent breakthrough result of Bowen, Kun and Sabok that enables one to obtain perfect matchings from fractional perfect matchings under suitable conditions.

ON THE SIZE, SPECTRAL RADIUS, DISTANCE SPECTRAL RADIUS AND FRACTIONAL MATCHINGS IN GRAPHS

AbstractWe first establish a lower bound on the size and spectral radius of a graph G to guarantee that G contains a fractional perfect matching. Then, we determine an upper bound on the distance spectral radius of a graph G to ensure that G has a fractional perfect matching. Furthermore, we construct some extremal graphs to show all the bounds are best possible.

Conditional fractional matching preclusion for burnt pancake graphs and pancake-like graphs

ABSTRACT The conditional fractional strong matching preclusion number of a graph G is the minimum size of F such that and G−F has neither a fractional perfect matching nor an isolated vertex. In this paper, we obtain the conditional fractional strong matching preclusion number for burnt pancake graphs and a subset of the class of pancake-like graphs.

The nullity of a graph with fractional matching number

Let G be a simple graph with n(G) vertices and e(G) edges. Denoted by η(G) and m⁎(G) the nullity and the fractional matching number of G, respectively. The dimension of cycle space of G is defined as c(G)=e(G)−n(G)+ω(G), where ω(G) denotes the number of connected components of G.In this paper, we prove that n(G)−2m⁎(G)≤η(G)≤n(G)−2m⁎(G)+2c(G) for a graph G, which improves the main results of Wang and Wong (2014) [19] and Ma and Fang (2019) [11], respectively. Furthermore, all graphs with nullity η(G)=n−2m⁎(G)+2c(G) are determined. We also prove that there is no graph with nullity η(G)=n−2m⁎(G)+2c(G)−1; and for fixed c(G), infinitely many connected graphs with nullity n−2m⁎(G)+2c(G)−k(0≤k≤2c(G),k≠1) are also constructed. As an application of the above results, we also prove that if G is nonsingular, then G has a fractional perfect matching.

A Note on Exact Minimum Degree Threshold for Fractional Perfect Matchings

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 \(k-1>d\ge k/2\), and let n be any integer with \(n\ge 2k(k-1)+1\). Then any n-vertex k-uniform hypergraph with minimum d-degree \(\delta _d(H)>{n-d\atopwithdelims ()k-d} -{n-d-(\lceil n/k\rceil -1)\atopwithdelims ()k-d}\) contains a fractional perfect matching. This lower bound on the minimum d-degree is best possible. We also determine the minimum d-degree threshold for 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 \(2k/5<d<k/2\)).

Fractional matching preclusion number of graphs

The 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.

On the global strong resilience of fault Hamiltonian graphs

The global strong resilience of G with respect to having a fractional perfect matching, also called FSMP number of G, is the minimum number of edges (or resp., edges and/or vertices) whose deletion results in a graph that has no fractional perfect matchings. A graph G is said to be f-fault Hamiltonian if there exists a Hamiltonian cycle in G−F for any set F of edges and/or vertices with |F|≤f. In this paper, we first give a sufficient condition, involving the independent number, to determine the FSMP number of (δ−2)-fault Hamiltonian graphs with minimum degree δ≥2, and then we can derive the FSMP number of some networks, which generalize some known results.

Fractional strong matching preclusion of some Cartesian product graphs

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 the Cartesian product of a graph and a cycle. As an application, the fractional strong matching preclusion number for torus networks is also obtained.

The fractional (strong) matching preclusion number of complete k-partite graph

The fractional (strong) matching preclusion number of a graph G, denoted by f(s)mp(G), is the minimum number of edges (and vertices) whose deletion results in a graph with no fractional perfect matching. Let Gn1,n2,…,nk be the complete k-partite graph, whose vertex set can be partitioned into k parts, each has ni(1≤i≤k) vertices and whose edge set contains all edges between two distinct parts. In this paper, we determine fmp(Gn1,n2,…,nk) and fsmp(Gn1,n2,…,nk).

Fractional Matching Preclusion for Data Center Networks

An edge subset [Formula: see text] of [Formula: see text] is a fractional matching preclusion set (FMP set for short) if [Formula: see text] has no fractional perfect matchings. The fractional matching preclusion number (FMP number for short) of [Formula: see text], denoted by [Formula: see text], is the minimum size of FMP sets of [Formula: see text]. A set [Formula: see text] of edges and vertices of [Formula: see text] is a fractional strong matching preclusion set (FSMP set for short) if [Formula: see text] has no fractional perfect matchings. The fractional strong matching preclusion number (FSMP number for short) of [Formula: see text], denoted by [Formula: see text], is the minimum size of FSMP sets of [Formula: see text]. Data center networks have been proposed for data centers as a server-centric interconnection network structure, which can support millions of servers with high network capacity by only using commodity switches. In this paper, we obtain the FMP number and the FSMP number for data center networks [Formula: see text], and show that [Formula: see text] for [Formula: see text], [Formula: see text] and [Formula: see text] for [Formula: see text], [Formula: see text]. In addition, all the optimal fractional strong matching preclusion sets of these graphs are categorized.

On the König–Hall–Egerváry theorem for multidimensional matrices and multipartite hypergraphs

One of possible interpretations of the well-known König–Hall–Egerváry theorem is a full characterization of all bipartite graphs extremal for fractional matchings of a given weight (or, equivalently, a characterization of (0,1)-matrices extremal for partial fractional diagonals of a given length). In this paper we initiate the study of d-partite d-uniform hypergraphs that are extremal for fractional perfect matchings (or, equivalently, d-dimensional (0,1)-matrices that are extremal for polydiagonals). For this purpose, we analyze similarities and differences between 2-dimensional and multidimensional cases and put forward a series of questions and conjectures on properties of multidimensional extremal matrices (extremal hypergraphs). We also prove these conjectures for several parameters and provide a number of supporting constructions and examples.

Conditional fractional matching preclusion of [formula omitted]-dimensional torus networks

The fractional matching preclusion number of a graph is the minimum number of edges whose deletion results in the remaining graph that has no fractional perfect matchings. For many networks, their optimal fractional matching preclusion sets are precisely those edges incident with a single vertex. The probability that all failures concentrate around a vertex is often small. To overcome the shortcoming, we consider the concept of conditional fractional matching preclusion, in which isolated vertices are not permitted in fault networks. We establish the conditional fractional matching preclusion numbers and all possible minimum conditional fractional matching preclusion sets for n-dimensional torus networks with n≥3.

Fractional matching number and spectral radius of nonnegative matrices of graphs

ABSTRACT A fractional matching of a graph G is a function f:E(G) → [0, 1] such that for any v ∈ V(G), where E G (v) = {e ∈ E(G): e is incident with v in G}. The fractional matching number of G is is a fractional matching of G}. For any real numbers a ≥ 0 and k ∈ (0, n), it is observed that if n = |V(G)| and , then . We determine a function φ(a, n, δ, k) and show that for a connected graph G with n = |V(G)|, , spectral radius λ 1(G) and complement , each of the following holds. If λ 1 (aD(G) + A(G)) < φ(a, n, δ, k), then If then As applications, we prove a relationship between μ f (G) and λ 1(aD(G) + A(G)) for a graph G. Furthermore, sufficient spectral conditions for a graph to have a fractional perfect matching are also obtained.

Fractional Matching Preclusion for Folded Petersen Cube Networks

Let F be an edge set and F′ a subset of edges and/or vertices of a graph G. Then F is a fractional matching preclusion(FMP) set (F′ is a fractional strong matching preclusion (FSMP) set) if G − F (G − F′) does not contain fractional perfect matching. The FMP(FSMP) number of G is the minimum size of FMP(FSMP) sets of G. The concept of matching preclusion was introduced by Brigham et al., as a measure of robustness in the event of edge failure in interconnection networks. An interconnection network of a larger MP number may be considered as more robust in the event of link failures. The problem of fractional matching preclusion is a generalization of matching preclusion. In this paper, we obtain the FMP and FSMP number for the folded Petersen cube networks. All the optimal fractional strong matching preclusion sets of these graphs are categorized.

Fractional matching preclusion of product networks

The matching preclusion number of a graph, introduced in [2] as a fault analysis, is the minimum number of edges whose deletion leaves a resulting graph that has neither perfect matchings nor almost perfect matchings. As a generalization, Liu and Liu [14] recently introduced the concept of fractional matching preclusion number. The fractional matching preclusion number of graph is the minimum number of edges whose deletion results in a graph that has no fractional perfect matching. If the sets of edges of the graph attaining the minimum are precisely those incident to a single vertex of minimum degree, we say such graph is fractional super matched. In this paper, the upper and lower bounds for the fractional matching preclusion number for Cartesian product, direct product, strong product, and lexicographic product are obtained, and we give sufficient conditions for such graphs to be fractional super matched.