Edge Decomposition Research Articles

Exact values for some unbalanced Zarankiewicz numbers

AbstractFor positive integers , , and , the Zarankiewicz number is defined to be the maximum number of edges in a bipartite graph with parts of sizes and that has no complete bipartite subgraph containing vertices in the part of size and vertices in the part of size . A simple argument shows that, for each , when . Here, for large , we determine the exact value of in almost all of the remaining cases where . We establish a new family of upper bounds on which complement a family already obtained by Roman. We then prove that the floor of the best of these bounds is almost always achieved. We also show that there are cases in which this floor cannot be achieved and others in which determining whether it is achieved is likely a very hard problem. Our results are proved by viewing the problem through the lens of linear hypergraphs and our constructions make use of existing results on edge decompositions of dense graphs.

Gregarious Decompositions of Complete Equipartite Graphs and Related Structures

In a finite mathematical structure with a given partition, a substructure is said to be gregarious if either it meets each partition class or it shares at most one element with each partition class. In this paper, we considered edge decompositions of graphs and hypergraphs into gregarious subgraphs and subhypergraphs. We mostly dealt with “complete equipartite” graphs and hypergraphs, where the vertex classes have the same size and precisely those edges or hyperedges of a fixed cardinality are present that do not contain more than one element from any class. Some related graph classes generated by product operations were also considered. The generalization to hypergraphs offers a wide open area for further research.

Spectrum of 3-uniform 6- and 9-cycle systems over [formula omitted

We consider edge decompositions of Kv(3)−I, the complete 3-uniform hypergraph of order v minus a set of v/3 mutually disjoint edges (1-factor). We prove that a decomposition into tight 6-cycles exists if and only if v≡0,3,6 (mod 12) and v≥6; and a decomposition into tight 9-cycles exists for all v≥9 divisible by 3. These results are complementary to the theorems of Akin et al. [Discrete Math. 345 (2022)] and Bunge et al. [Australas. J. Combin. 80 (2021)] who settled the case of Kv(3).

Generalizations of some Nordhaus–Gaddum‐type results on spectral radius

AbstractLet be a simple graph and the spectral radius of . For , a ‐edge decomposition is spanning subgraphs such that their edge sets form a ‐partition of the edge set of . In this paper, we obtain some sharp lower and upper bounds for in terms of the clique number of and the size of , and discuss what ‐edge decomposition can maximize when is a complete graph. These generalize some Nordhaus–Gaddum‐type results on spectral radius for , due to Nosal, Hong and Shu, and Nikiforov.

Every subcubic multigraph is (1,27) $(1,{2}^{7})$‐packing edge‐colorable

AbstractFor a nondecreasing sequence of positive integers, an ‐packing edge‐coloring of a graph is a decomposition of edges of into disjoint sets such that for each the distance between any two distinct edges is at least . The notion of ‐packing edge‐coloring was first generalized by Gastineau and Togni from its vertex counterpart. They showed that there are subcubic graphs that are not ‐packing (abbreviated to ‐packing) edge‐colorable and asked the question whether every subcubic graph is ‐packing edge‐colorable. Very recently, Hocquard, Lajou, and Lužar showed that every subcubic graph is ‐packing edge‐colorable and every 3‐edge‐colorable subcubic graph is ‐packing edge‐colorable. Furthermore, they also conjectured that every subcubic graph is ‐packing edge‐colorable.In this paper, we confirm the conjecture of Hocquard, Lajou, and Lužar, and extend it to multigraphs.

Impact of Polymer-Assisted Epitaxial Graphene Growth on Various Types of SiC Substrates.

The growth parameters for epitaxial growth of graphene on silicon carbide (SiC) have been the focus of research over the past few years. However, besides the standard growth parameters, the influence of the substrate pretreatment and properties of the underlying SiC wafer are critical parameters for optimizing the quality of monolayer graphene on SiC. In this systematic study, we show how the surface properties and the pretreatment determine the quality of monolayer graphene using polymer-assisted sublimation growth (PASG) on SiC. Using the spin-on deposition technique of PASG, several polymer concentrations have been investigated to understand the influence of the polymer content on the final monolayer coverage using wafers of different miscut angles and different polytypes. Confocal laser scanning microscopy (CLSM), atomic force microscopy (AFM), Raman spectroscopy, and scanning electron microscopy (SEM) were used to characterize these films. The results show that, even for SiC substrates with high miscut angles, high-quality graphene is obtained when an appropriate polymer concentration is applied. This is in excellent agreement with the model understanding that an insufficient carbon supply from SiC step edge decomposition can be compensated by additionally providing carbon from a polymer source. The described methods make the PASG spin-on deposition technique more convenient for commercial use.

Edge decompositions and rooted packings of graphs

Let H be a fixed graph. In this paper we consider the problem of edge decomposition of a graph into subgraphs isomorphic to H or 2K2 (a 2-edge matching). We give a partial classification of the problems of existence of such decomposition according to the computational complexity. More specifically, for some large class of graphs H we show that this problem is polynomial time solvable and for some other large class of graphs it is NP-complete. These results can be viewed as some edge decomposition analogs of a result by Loebl and Poljak who classified according to the computational complexity the problem of existence of a graph factor with components isomorphic to H or K2. In the proofs of our results we apply so-called rooted packings into graphs which are mutual generalizations of both edge decompositions and factors of graphs.

On the decomposition of circulant graphs using algorithmic approaches

Many structural models in chemistry, biology, computer science, sociology, and operations research can be analyzed using graph theory. Some examples of these structure models are species movement between regions, molecular bonds, shortest spanning trees, development of computer algorithms. This paper introduces the edge decomposition of circulant graphs with 2n vertices by different graph classes. These circulant graphs are denoted as C2n,n, where n is the degree of the graph C2n,n. We propose two algorithmic approaches for constructing edge decomposition of C2n,n. With aid of the proposed algorithmic approaches, we construct paths decompositions, trees decompositions, disjoint unions of cycles and trees decompositions, and complete bipartite graphs decompositions of C2n,n. In the end, a recursive construction of an edge decomposition, based on orthogonal edge decompositions, is proposed. This helps in using the mutually orthogonal decompositions of the circulant graphs C2n,n in several applications, such as the design of experiments and binary codes generation.

An Evans-Style Result for Block Designs

For positive integers $n$ and $k$ with $n \geq k$, an $(n,k,1)$-design is a pair $(V, \mathcal{B})$ where $V$ is a set of $n$ points and $\mathcal{B}$ is a collection of $k$-subsets of $V$ called blocks such that each pair of points occur together in exactly one block. If we weaken this condition to demand only that each pair of points occur together in at most one block, then the resulting object is a partial $(n,k,1)$-design. A completion of a partial $(n,k,1)$-design $(V,\mathcal{A})$ is a (complete) $(n,k,1)$-design $(V,\mathcal{B})$ such that $\mathcal{A} \subseteq \mathcal{B}$. Here, for all sufficiently large $n$, we determine exactly the minimum number of blocks in an uncompletable partial $(n,k,1)$-design. This result is reminiscent of Evans' now-proved conjecture on completions of partial latin squares. We also prove some related results concerning edge decompositions of almost complete graphs into copies of $K_k$.

A novel application on mutually orthogonal graph squares and graph-orthogonal arrays

<abstract> <p>Security of personal information has become a major concern due to the increasing use of the Internet by individuals in the digital world. The main purpose here is to prevent an unauthorized person from gaining access to confidential information. The solution to such a problem is by authentication of users. Authentication has a very important role in achieving security. Mutually orthogonal graph squares (MOGS) are considered the generalization of mutually orthogonal Latin squares (MOLS). Also, MOGS are generated from edge decompositions of complete bipartite graphs by isomorphic graphs. Graph-orthogonal arrays can be constructed by MOGS. In this paper, graph-orthogonal arrays are used for constructing authentication codes. These arrays are the encoding matrices of authentication tags. We introduce the concepts and basic theorems of MOGS, graph-orthogonal arrays, and authentication codes. After constructing graph-orthogonal arrays by MOGS, then there is an established mapping between graph-orthogonal arrays and message set. This manages us to construct perfect non-splitting and splitting Cartesian authentication codes. In both cases, we calculate the probabilities of successful impersonation attacks and substitution attacks. Besides that, the performance of constructed non-splitting and splitting authentication codes is analyzed. In the end, optimal authentication codes and secure authentication codes are constructed.</p> </abstract>

Privacy-Preserving Average Consensus Via Edge Decomposition

Average consensus problem in multi-agent systems has been an active topic allowing multiple agents to agree on the average of their initial values through local information interaction. However, the explicit sharing of state variables may lead to privacy disclosure problem. In this letter, a new approach is proposed to protect the initial state information of agents while ensuring convergence to the correct average consensus value. The core idea to achieve these goals is based on edge decomposition strategy. Agent divides its each connected edge into two edges, one edge is still connected to the original node, and the other edge is connected to a virtual node. Different from the existing state decomposition approach, our method changes from considering the nodes in network to considering the edges, which provides a new perspective for privacy protection. The correctness analysis and privacy analysis of the method are provided subsequently. Finally, numerical simulations are given to verify the effectiveness of our approach.

Classification and Automatic Feature-Based Extraction Approach for Cylindrical and Milling Parts

This paper reports classification and automatic extraction of various cylindrical and milling features in conventional machining process parts. In this work, various algorithms like hole recognition algorithm (HRA) and milling feature recognition algorithm (MFRA) have been used for identification of different cylindrical and milling features. A cylindrical feature is identified based on specific logical rules, and milling feature is identified based on the concept of concave decomposition of edges. In-house developed JAVA program is used to write algorithm, and then validation of the algorithm is done through two case studies. The HRA and MFRA algorithms extract the cylindrical features (through holes, blind holes, taper holes, and bosses) and milling features (slot, blind slot, step, blind step, pockets) precisely. The current work is well suitable to extract the features in conventional machining parts and thereby improve the downstream applications likes process planning, CAPP, CAM, etc.

In-situ investigation on the thermal decomposition of van der Waals MoO3

The van der Waals α-MoO3 is one of the most attracted two-dimensional metal oxides in opto-electronics Thermal stability is one of the key properties in device application. Here, the thermal decomposition of individual α-MoO3 nanosheet was investigated using in-situ microscope and Raman spectroscopy. It’s found that the nanosheet decomposes above 400 °C. Three decomposition behaviors, including void decomposition, edge decomposition and layer-by-layer decomposition were found. The Oa atom is the most unstable oxygen atom that causes a preferential decomposition along the [0 0 1] direction. Thermal stress accumulated in the [1 0 0] direction leads to a non-linear temperature dependent Raman shifting.

Long path and cycle decompositions of even hypercubes

We consider edge decompositions of the n-dimensional hypercube Qn into isomorphic copies of a given graph H. While a number of results are known about decomposing Qn into graphs from various classes, the simplest cases of paths and cycles of a given length are far from being understood. A conjecture of Erde asserts that if n is even, ℓ<2n and ℓ divides the number of edges of Qn, then the path of length ℓ decomposes Qn. Tapadia et al. proved that any path of length 2mn, where 2m<n, satisfying these conditions decomposes Qn. Here, we make progress toward resolving Erde’s conjecture by showing that cycles of certain lengths up to 2n+1∕n decompose Qn. As a consequence, we show that Qn can be decomposed into copies of any path of length at most 2n∕n dividing the number of edges of Qn, thereby settling Erde’s conjecture up to a linear factor.

Uniformly Resolvable Decompositions of Kv-I into n-Cycles and n-Stars, for Even n

If X is a connected graph, then an X-factor of a larger graph is a spanning subgraph in which all of its components are isomorphic to X. Given a set Γ of pairwise non-isomorphic graphs, a uniformly resolvable Γ-decomposition of a graph G is an edge decomposition of G into X-factors for some graph X∈Γ. In this article we completely solve the existence problem for decompositions of Kv-I into Cn-factors and K1,n-factors in the case when n is even.

Sharp bounds for decomposing graphs into edges and triangles

AbstractFor a real constant α, let $\pi _3^\alpha (G)$ be the minimum of twice the number of K2’s plus α times the number of K3’s over all edge decompositions of G into copies of K2 and K3, where Kr denotes the complete graph on r vertices. Let $\pi _3^\alpha (n)$ be the maximum of $\pi _3^\alpha (G)$ over all graphs G with n vertices.The extremal function $\pi _3^3(n)$ was first studied by Győri and Tuza (Studia Sci. Math. Hungar.22 (1987) 315–320). In recent progress on this problem, Král’, Lidický, Martins and Pehova (Combin. Probab. Comput.28 (2019) 465–472) proved via flag algebras that$\pi _3^3(n) \le (1/2 + o(1)){n^2}$. We extend their result by determining the exact value of $\pi _3^\alpha (n)$ and the set of extremal graphs for all α and sufficiently large n. In particular, we show for α = 3 that Kn and the complete bipartite graph ${K_{\lfloor n/2 \rfloor,\lceil n/2 \rceil }}$ are the only possible extremal examples for large n.

Decompositions of complete multigraphs into stars of varying sizes

In 1979 Tarsi showed that an edge decomposition of a complete multigraph into stars of size m exists whenever some obvious necessary conditions hold. In 1992 Lonc gave necessary and sufficient conditions for the existence of an edge decomposition of a (simple) complete graph into stars of sizes m1,…,mt. We show that the general problem of when a complete multigraph admits a decomposition into stars of sizes m1,…,mt is NP-complete, but that it becomes tractable if we place a strong enough upper bound on max⁡(m1,…,mt). We determine the upper bound at which this transition occurs. Along the way we also give a characterisation of when an arbitrary multigraph can be decomposed into stars of sizes m1,…,mt with specified centres, and a generalisation of Landau's theorem on tournaments.

Decomposing tournaments into paths

We consider a generalisation of Kelly's conjecture which is due to Alspach, Mason, and Pullman from 1976. Kelly's conjecture states that every regular tournament has an edge decomposition into Hamilton cycles, and this was proved by K\"uhn and Osthus for large tournaments. The conjecture of Alspach, Mason, and Pullman asks for the minimum number of paths needed in a path decomposition of a general tournament $T$. There is a natural lower bound for this number in terms of the degree sequence of $T$ and it is conjectured that this bound is correct for tournaments of even order. Almost all cases of the conjecture are open and we prove many of them.

Edge decomposition of connected claw-free cubic graphs

It was conjectured by Hoffmann–Ostenhof that the edge set of every connected cubic graph can be decomposed into a spanning tree, a matching and a family of disjoint cycles. In this paper, we show that the conjecture is true for connected claw-free cubic graphs, and, furthermore, any edge not contained any triangle appears only on the tree or on the matching. Then we show that the edge set of every connected cubic graph (except for K4 and K3,3) can be decomposed into a spanning tree and a family of disjoint paths of length at most 2.

Decomposition of Core-Loss EEL Edges into a Linear Combination of Refined ‘Pure’ Spectra

Decomposition of Core-Loss EEL Edges into a Linear Combination of Refined ‘Pure’ Spectra