Partition Of Vertex Set Research Articles

Tiling with monochromatic bipartite graphs of bounded maximum degree

AbstractWe prove that for any , there exists a constant such that the following is true. Let be an infinite sequence of bipartite graphs such that and hold for all . Then, in any ‐edge‐coloured complete graph , there is a collection of at most monochromatic subgraphs, each of which is isomorphic to an element of , whose vertex sets partition . This proves a conjecture of Corsten and Mendonça in a strong form and generalises results on the multi‐colour Ramsey numbers of bounded‐degree bipartite graphs. It also settles the bipartite case of a general conjecture of Grinshpun and Sárközy.

Topological structures induced by chromatic partitioning of vertex set of graphs

This paper presents a method of constructing topologies on vertex set of a graph G induced by chromatic partition of vertex set of the graph. It introduces colour lower approximation and colour upper approximation of vertex induced subgraphs and acquaints the open and closed sets of the topology generated by chromatic partition on the vertex set of graphs. It explores some of the properties of colour lower approximation and colour upper approximation of vertex induced subgraphs. It also establishes some new subgraphs based on the colour lower approximation and colour upper approximation and some of their properties have been studied.

On the existence of k $k$‐cycle semiframes for even k $k$

AbstractA ‐semiframe of type is a ‐group divisible design of type in which is the vertex set, is the group set, and the set of ‐cycles can be written as a disjoint union where is partitioned into parallel classes on and is partitioned into holey parallel classes, each parallel class or holey parallel class being a set of vertex disjoint cycles whose vertex sets partition or for some . In this paper, we almost completely solve the existence of a ‐semiframe of type for all and a ‐semiframe of type for all and .

Some results on generalized self-complementary graphs

Let [Formula: see text] be a partition of vertex set [Formula: see text] of order [Formula: see text] of a graph [Formula: see text]. The [Formula: see text]-complement of [Formula: see text] denoted by [Formula: see text] is defined as for all [Formula: see text] and [Formula: see text] in [Formula: see text], [Formula: see text], remove the edges between [Formula: see text] and [Formula: see text] in [Formula: see text] and add the edges between [Formula: see text] and [Formula: see text] which are not in [Formula: see text]. The graph [Formula: see text] is called [Formula: see text]-self-complementary if [Formula: see text]. For a graph [Formula: see text], [Formula: see text]-complement of [Formula: see text] denoted by [Formula: see text] is defined as for each [Formula: see text] remove the edges of [Formula: see text] inside [Formula: see text] and add the edges of [Formula: see text] by joining the vertices of [Formula: see text]. The graph [Formula: see text] is called [Formula: see text]-self-complementary if [Formula: see text] for some partition [Formula: see text] of order [Formula: see text]. In this paper, we determine generalized self-complementary graphs of forest, double star and unicyclic graphs.

On joins of a clique and a co-clique as star complements in regular graphs

In this paper we consider r-regular graphs G that admit the vertex set partition such that one of the induced subgraphs is the join of an s-vertex clique and a t-vertex co-clique and represents a star complement for an eigenvalue \(\mu \) of G. The cases in which one of the parameters s, t is less than 2 or \(\mu =r\) are already resolved. It is conjectured in Wang et al. (Linear Algebra Appl 579:302–319, 2019) that if \(s, t\ge 2\) and \(\mu \ne r\), then \(\mu =-2, t=2\) and \(G=\overline{(s+1)K_2}\). For \(\mu =-t\) we verify this conjecture to be true. We further study the case in which \(\mu \ne -t\) and confirm the conjecture provided \(t^2-4\mu ^2t-4\mu ^3=0\). For the remaining possibility we determine the structure of a putative counterexample and relate its existence to the existence of a particular 2-class block design. It occurs that the smallest counterexample would have 1265 vertices.

Some properties of generalized complements of a graph

Let $P=\{V_{1},V_{2},\cdots,V_{k}\}$ be a partition of vertex set $V$ of $G$. The $k-$complement of $G$ denoted by $G_{k}^{P}$ is defined as follows: for all $V_{i}$ and $V_{j}$ in $P$, $i\neq j$, remove the edges between $V_{i}$ and $V_{j}$ and add edges between $V_{i}$ and $V_{j}$ which are not in $G$. The graph $G$ is k-self complementary with respect to $P$ if $G_{k}^{P}\cong G$. The k(i)-complement $G_{k(i)}^{P}$ of a graph $G$ with respect to $P$ is defined as follows: for all $V_{r}\in P$, remove edges inside $V_{r}$ and add edges which are not in $V_{r}$. In this paper we provide sufficient conditions for $G_{k}^{P}$ and $G_{k(i)}^{P}$ to be disconnected, regular, line preserving and Eulerian.

A decomposition of signed graphs with two eigenvalues

In this study we consider connected signed graphs with 2 eigenvalues that admit a vertex set partition such that the induced signed graphs also have 2 eigenvalues, each. We derive some spectral characterizations, along with examples supported by additional theoretical results. We also prove an inequality that is a fundamental ingredient for the resolution of the Sensitivity Conjecture.

Upper bound for radio [formula omitted]-chromatic number of graphs in connection with partition of vertex set

For a simple connected graph G=(V(G),E(G)) and a positive integer k with 1⩽k⩽diam(G), a radio k-coloring of G is a mapping f:V(G)→{0,1,2,…} such that |f(u)−f(v)|⩾k+1−d(u,v) holds for each pair of distinct vertices u and v of G, where diam(G) is the diameter of G and d(u,v) is the distance between u and v in G. The span of a radio k-coloring f is the number maxu∈V(G)f(u). The main aim of the radio k-coloring problem is to determine the minimum span of a radio k-coloring of G, denoted by rck(G). Due to NP-hardness of this problem, people are finding lower bounds for rck(.) for particular families of graphs or general graphs G. In this article, we concentrate on finding upper bounds of radio k-chromatic number for general graphs and in consequence a coloring scheme depending on a partition of the vertex set V(G). We apply these bounds to cycle graph Cn and hypercube Qn and show that it is sharp when k is close to the diameter of these graphs.

Laplacian energy of generalized complements of a graph

Let P = {V1, V2, V3, . . . , Vk} be a partition of vertex set V (G) of order k ≥ 2. For all Vi and Vj in P, ≠ 6= j, remove the edges between Vi and Vj in graph G and add the edges between Vi and Vj which are not in G. The graph GPk thus obtained is called the k-complement of graph G with respect to a partition P. For each set Vr in P, remove the edges of graph G inside Vr and add the edges of G (the complement of G) joining the vertices of Vr. The graph GPk (i) thus obtained is called the k(i)-complement of graph G with respect to a partition P. In this paper, we study Laplacian energy of generalized complements of some families of graph. An effort is made to throw some light on showing variation in Laplacian energy due to changes in a partition of the graph.

The Smallest Uniform Color-Bounded Hypergraphs Which are One-Realizations of a Given Set

A color-bounded hypergraph is a hypergraph with vertex set X and edge set \({\mathcal {E}}=\{E_1,E_2,\dots ,E_m\}\), together with integers \(s_i\) and \(t_i\) (\(1\le s_i\le t_i\le |E_i|\)) for \(i=1,2,\ldots ,m\). A vertex coloring \(\varphi \) is proper if the number of colors occurring in edge \(E_i\) satisfies \(s_i\le |\varphi (E_i)|\le t_i\), for every \(1\le i\le m\). If \(s_i=s\) and \(t_i=t\) for all i, we simply denote the color-bounded hypergraph by \({\mathcal {H}}=(X, {\mathcal {E}},s,t)\). A set of positive integers \(\Phi (\mathcal {H})\) is called feasible, if it consists of all k for which there exists a proper coloring of \(\mathcal {H}\) using precisely k colors. Chromatic spectrum of a hypergraph \(\mathcal {H}\) is a vector with each entry \(r_k\) equal to the number of partitions of vertex set induced by all proper colorings using k colors. Let S be a finite set of positive integers. A color-bounded hypergraph is a one-realization of S if its feasible set is S and each entry of its chromatic spectrum is either 0 or 1. In this paper, we determine the minimum number of vertices of r-uniform color-bounded hypergraphs \({\mathcal {H}}=(X, {\mathcal {E}},2,t)\) which are one-realizations of S for the case when \(\lceil \frac{r}{2}\rceil <t\le r-2\) and \(\max (S)\ge \frac{3r}{2}\).

The adjacency matrix of a graph as a data table: a geometric perspective

In this paper we continue a research project concerning the study of a graph from the perspective of granular computation. To be more specific, we interpret the adjacency matrix of any simple undirected graph G in terms of data information table, which is one of the most studied structures in database theory. Granular computing (abbreviated GrC) is a well-developed research field in applied and theoretical information sciences; nevertheless, in this paper we address our efforts toward a purely mathematical development of the link between GrC and graph theory. From this perspective, the well-studied notion of indiscernibility relation in GrC becomes a symmetry relation with respect to a given vertex subset in graph theory; therefore, the investigation of this symmetry relation turns out to be the main object of study in this paper. In detail, we study a simple undirected graph G by assuming a generic vertex subset W as reference system with respect to which examine the symmetry of all vertex subsets of G. The change of perspective from G without reference system to the pair (G, W) is similar to what occurs in the transition from an affine space to a vector space. We interpret the symmetry blocks in the reference system (G, W) as particular equivalence classes of vertices in G, and we study the geometric properties of all reference systems (G, W), when W runs over all vertex subsets of G. We also introduce three hypergraph models and a vertex set partition lattice associated to G, by taking as general models of reference several classical notions of GrC. For all these constructions, we provide a geometric characterization and we determine their structure for basic graph families. Finally, we apply a wide part of our work to study the important case of the Petersen graph.

Extremal values and bounds for the zero forcing number

A set Z of vertices of a graph G is a zero forcing set of G if iteratively adding to Z vertices from V(G)∖Z that are the unique neighbor in V(G)∖Z of some vertex in Z, results in the entire vertex set V(G) of G. The zero forcing number Z(G) of G is the minimum cardinality of a zero forcing set of G.Amos et al. (2015) proved Z(G)≤((Δ−2)n+2)/(Δ−1) for a connected graph G of order n and maximum degree Δ≥2. Verifying their conjecture, we show that Cn, Kn, and KΔ,Δ are the only extremal graphs for this inequality. Confirming a conjecture of Davila and Kenter [5], we show that Z(G)≥2δ−2 for every triangle-free graph G of minimum degree δ≥2. It is known that Z(G)≥P(G) for every graph G where P(G) is the minimum number of induced paths in G whose vertex sets partition V(G). We study the class of graphs G for which every induced subgraph H of G satisfies Z(H)=P(H).

Strategic balance in graphs

For a given graph G, a nonempty subset S contained in V(G) is an alliance iff for each vertex v∈S there are at least as many vertices from the closed neighbourhood of v in S as in V(G)−S. An alliance is global if it is also a dominating set of G. The alliance partition number of G was defined in Hedetniemi et al. (2004) to be the maximum number of sets in a partition of V(G) such that each set is an alliance. Similarly, in Eroh and Gera (2008) the global alliance partition number is defined for global alliances, where the authors studied the problem for (binary) trees.In the paper we introduce a new concept of strategic balance in graphs: for a given graph G, determine whether there is a partition of vertex set V(G) into three subsets N, S and I such that both N and S are global alliances. We give a survey of its general properties, e.g., showing that a graph G has a strategic balance iff its global alliance partition number equals at least 2. We construct a linear time algorithm for solving the problem in trees (thus giving an answer to the open question stated in Eroh and Gera (2008)) and studied this problem for many classes of graphs: paths, cycles, wheels, stars, complete graphs and complete k-partite graphs. Moreover, we prove that this problem is NP-complete for graphs with a degree bounded by 4 and state an open question regarding subcubic graphs.

The generalized split probe problem

A generalized split (k,l) partition is a vertex set partition into at most k independent sets and l cliques. We prove that the (2, 1) partitioned probe problem is in P whereas the (2, 2) partitioned probe is NP-complete. The full complexity dichotomy into polynomial time and NP-complete for the class of generalized split partitioned probe problems establishes (2, 2) as the first NP-complete self-complementary partitioned probe problem, and answers negatively the PGC conjecture by finding a polynomial time recognition problem whose partitioned probe version is NP-complete.

Formula omitted] vertex-set partition into nonempty parts

A graph is 2 K 2 -partitionable if its vertex set can be partitioned into four nonempty parts A , B , C , D such that each vertex of A is adjacent to each vertex of B , and each vertex of C is adjacent to each vertex of D . Determining whether an arbitrary graph is 2 K 2 -partitionable is the only vertex-set partition problem into four nonempty parts according to external constraints whose computational complexity is open. We establish that the 2 K 2 -partition problem parameterized by minimum degree is fixed-parameter tractable. We also show that for C 4 -free graphs, circular-arc graphs, spiders, P 4 -sparse graphs, and bipartite graphs the 2 K 2 -partition problem can be solved in polynomial time.

Factors in random graphs

AbstractLet H be a fixed graph on v vertices. For an n‐vertex graph G with n divisible by v, an H‐factor of G is a collection of n/v copies of H whose vertex sets partition V (G).In this work, we consider the threshold thH(n) of the property that an Erdős‐Rényi random graph (on n points) contains an H‐factor. Our results determine thH(n) for all strictly balanced H.The method here extends with no difficulty to hypergraphs. As a corollary, we obtain the threshold for a perfect matching in random k‐uniform hypergraph, solving the well‐known “Shamir's problem.” © 2008 Wiley Periodicals, Inc. Random Struct. Alg., 2008

2K2 vertex-set partition into nonempty parts

Abstract A graph is 2K2-partitionable if its vertex set can be partitioned into four nonempty parts A, B, C, D such that each vertex of A is adjacent to each vertex of B, and each vertex of C is adjacent to each vertex of D. Determining whether an arbitrary graph is 2K2-partitionable is the only vertex-set partition problem into four nonempty parts according to external constraints whose computational complexity is open. We show that for C4-free graphs, circular-arc graphs, spiders, P4-sparse graphs, and bipartite graphs the 2K2-partition problem can be solved in polynomial time.

Asymptotic Existence of Resolvable Graph Designs

AbstractLet v ≥ k ≥ 1 and λ ≥ 0 be integers. A block design BD(v, k, λ) is a collection of k-subsets of a v-set X in which every unordered pair of elements from X is contained in exactly λ elements of . More generally, for a fixed simple graph G, a graph design GD(v, G, λ) is a collection of graphs isomorphic to G with vertices in X such that every unordered pair of elements from X is an edge of exactly λ elements of . A famous result of Wilson says that for a fixed G and λ, there exists a GD(v, G, λ) for all sufficiently large v satisfying certain necessary conditions. A block (graph) design as above is resolvable if can be partitioned into partitions of (graphs whose vertex sets partition) X. Lu has shown asymptotic existence in v of resolvable BD(v, k, λ), yet for over twenty years the analogous problem for resolvable GD(v, G, λ) has remained open. In this paper, we settle asymptotic existence of resolvable graph designs.

Partitioning Complete Bipartite Graphs by Monochromatic Cycles

For every positive integer r there exists a constant C r depending only on r such that for every colouring of the edges of the complete bipartite graph K n , n with r colours, there exists a set of at most C r monochromatic cycles whose vertex sets partition the vertex set of K n , n . This answers a question raised by Erdős, Gyárfás, and Pyber.

Partitioning by Monochromatic Trees

Any r -edge-coloured n -vertex complete graph K n contains at most r monochromatic trees, all of different colours, whose vertex sets partition the vertex set of K n , provided n ⩾3 r 4 r ! (1−1/ r ) 3(1− r ) log r . This comes close to proving, for large n , a conjecture of Erdo&#x030B;s, Gyárfás, and Pyber, which states that r −1 trees suffice for all n .