Complete K-partite Graph Research Articles

Packing 4-Partite Tree into Complete 4-Partite Graph

For graphs G and H, an embedding of G into H is an injection [see formula in PDF] such that [see formula in PDF] whenever [see formula in PDF]. A packing of p graphs [see formula in PDF] into [see formula in PDF]is a p-tuple [see formula in PDF] such that, for [see formula in PDF], [see formula in PDF] is an embedding of [see formula in PDF] into H and the p sets [see formula in PDF] are mutually disjoint. Motivated by the "Tree Packing Conjecture" made by Gy[see formula in PDF]rf[see formula in PDF]s and Lehel, Wang Hong conjectured that for each k-partite tree, there is a packing of two copies of [see formula in PDF] into a complete k-partite graph [see formula in PDF], where [see formula in PDF]. In this paper, we confirm this conjecture for [see formula in PDF].

On the existence of cycles with restrictions in the color transitions in edge-colored complete graphs

Consider the following edge-coloring of a graph G. Let H be a graph possibly with loops, an H-coloring of a graph G is defined as a function c:E(G)→V(H).\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$c : E(G) \\rightarrow V(H).$$\\end{document} We will say that G is an H-colored graph whenever we are taking a fixed H-coloring of G. A cycle (x0,x1,…,xn,x0),\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$(x_0,x_1,\\ldots ,x_n,x_0),$$\\end{document} in an H-colored graph, is an H-cycle if and only if (c(x0x1),c(x1x2),…,c(xnx0),\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$(c(x_0x_1),c(x_1x_2),\\ldots , c(x_nx_0),$$\\end{document}c(x0x1))\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$c(x_0x_1))$$\\end{document} is a walk in H. Notice that the graph H determines what color transitions are allowed in a cycle in order to be an H-cycle, in particular, when H is a complete graph without loops, every H-cycle is a properly colored cycle. In this paper, we give conditions on an H-colored complete graph G, with local restrictions, implying that every vertex of G is contained in an H-cycle of length at least 5. As a consequence, we obtain a previous result about properly colored cycles. Finally, we show an infinite family of H-colored complete graphs fulfilling the conditions of the main theorem, where the graph H is not a complete k-partite graph for any k in N.\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$${\\mathbb {N}}.$$\\end{document}

The k-Ramsey number of two five cycles

Given any two graphs F and H, the Ramsey number R(F, H) is defined as the smallest positive integer n such that every red-blue coloring of the edges of the complete graph Kn of order n, there will be a subgraph of Kn isomorphic to F whose edges are all colored red (a red F) or a subgraph of Kn isomorphic to H whose edges are all colored blue (a blue H). If F and H are bipartite graphs, then the k-Ramsey number R k ( F , H ) is defined as the smallest positive integer n such that for any red-blue coloring of the edges of the complete k-partite graph of order n in which each partite set is of order ⌊ n k ⌋ or ⌈ n k ⌉ there will be a subgraph isomorphic to F whose edges are all colored red (a red F) or a subgraph isomorphic to H whose edges are all colored blue (a blue H). Andrews, Chartrand, Lumduanhom and Zhang found the k-Ramsey number R k ( F , H ) for F = H = C 4 , and for F = K 1 , t and H = K 1 , s where s , t ≥ 2 . We continue their work by investigating the case where the graphs F and H are both C 5.

On decomposing multigraphs into locally irregular submultigraphs

A locally irregular multigraph is a multigraph whose adjacent vertices have distinct degrees. The locally irregular edge coloring is an edge coloring of a multigraph G such that every color induces a locally irregular submultigraph of G. We say that a multigraph G is locally irregular colorable if it admits a locally irregular edge coloring and we denote by lir(G) the locally irregular chromatic index of G, which is the smallest number of colors required in a locally irregular edge coloring of a locally irregular colorable multigraph G. We conjecture that for every connected graph G, which is not isomorphic to K2, the multigraph 2G obtained from G by doubling each edge admits lir(2G)≤2. This concept is closely related to the well known 1-2-3 Conjecture, Local Irregularity Conjecture, (2, 2) Conjecture and other similar problems concerning edge colorings. We show this conjecture holds for graph classes like paths, cycles, wheels, complete graphs, complete k-partite graphs and bipartite graphs. We also prove the general bound for locally irregular chromatic index for all 2-multigraphs using our result for bipartite graphs.

On distance Laplacian spectral determination of complete multipartite graphs

The (distance) Laplacian spectrum is said to determine a graph Γ if there is no non-isomorphic graph whose (distance) Laplacian spectra is same as that of Γ. Aouchiche et al. [3] proved that “complete k-partite graph is determined by its distance Laplacian spectrum”. This result is not true. In this paper, we determine the correct result on this. Further motivated by this, the distance Laplacian spectral determination of complete k-partite graph with edge addition is studied and at last it is shown that the graphs whose complements are disconnected and determined by their Laplacian spectra are also determined by their distance Laplacian spectra.

Sequential Clique Optimization for Unsupervised and Weakly Supervised Video Object Segmentation

A novel video object segmentation algorithm, which segments out multiple objects in a video sequence in unsupervised or weakly supervised manners, is proposed in this work. First, we match visually important object instances to construct salient object tracks through a video sequence without any user supervision. We formulate this matching process as the problem to find maximal weight cliques in a complete k-partite graph and develop the sequential clique optimization algorithm to determine the cliques efficiently. Then, we convert the resultant salient object tracks into object segmentation results and refine them based on Markov random field optimization. Second, we adapt the sequential clique optimization algorithm to perform weakly supervised video object segmentation. To this end, we develop a sparse-to-dense network to convert the point cliques into segmentation results. The experimental results demonstrate that the proposed algorithm provides comparable or better performances than recent state-of-the-art VOS algorithms.

The Strong Equitable Vertex 1-Arboricity of Complete Bipartite Graphs and Balanced Complete k-Partite Graphs

An equitable k-coloring of a graph G is a proper k-coloring of G such that the sizes of any two color classes differ by at most one. An equitable (q,r)-tree-coloring of a graph G is an equitable q-coloring of G such that the subgraph induced by each color class is a forest of maximum degree at most r. Let the strong equitable vertex r-arboricity of a graph G, denoted by var≡(G), be the minimum p such that G has an equitable (q,r)-tree-coloring for every q≥p. The values of va1≡(Kn,n) were investigated by Tao and Lin and Wu, Zhang, and Li where exact values of va1≡(Kn,n) were found in some special cases. In this paper, we extend their results by giving the exact values of va1≡(Kn,n) for all cases. In the process, we introduce a new function related to an equitable coloring and obtain a more general result by determining the exact value of each va1≡(Km,n) and va1≡(G) where G is a balanced complete k-partite graph Kn,…,n. Both complete bipartite graphs Km,n and balanced complete k-partite graphs Kn,…,n are symmetry in several aspects and also studied broadly. For the other aspect of symmetry, by the definition of equitable k-coloring of graphs, in a specific case that the number of colors divides the number of vertices of graph, we can say that the graph is a balanced k-partite graph.

Eccentricity energy change of complete multipartite graphs due to edge deletion

Abstract The eccentricity matrix ɛ(G) of a graph G is obtained from the distance matrix of G by retaining the largest distances in each row and each column, and leaving zeros in the remaining ones. The eccentricity energy of G is sum of the absolute values of the eigenvalues of ɛ(G). Although the eccentricity matrices of graphs are closely related to the distance matrices of graphs, a number of properties of eccentricity matrices are substantially different from those of the distance matrices. The change in eccentricity energy of a graph due to an edge deletion is one such property. In this article, we give examples of graphs for which the eccentricity energy increase (resp., decrease) but the distance energy decrease (resp., increase) due to an edge deletion. Also, we prove that the eccentricity energy of the complete k-partite graph Kn 1, ... , nk with k ≥ 2 and ni ≥ 2, increases due to an edge deletion.

A Cvetković-type Theorem for coloring of digraphs

In 1972, Cvetković proved that if G is an n-vertex simple graph with the chromatic number k, then its spectral radius is at most the spectral radius of the n-vertex balanced complete k-partite graph. In this paper, we analyze the characteristic polynomial of a digraph D to prove a tight upper bound for the spectral radius of D in terms of the number of vertices and the chromatic number of D; we also characterize when equality holds. This provides a simple proof of a result by Lin and Shu [6].

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).

The Minimum Stretch Spanning Tree Problem for Typical Graphs

With applications in communication networks, the minimum stretch spanning tree problem is to find a spanning tree T of a graph G such that the maximum distance in T between two adjacent vertices is minimized. The problem has been proved NP-hard and fixed-parameter polynomial algorithms have been obtained for some special families of graphs. In this paper, we concentrate on the optimality characterizations for typical classes of graphs. We determine the exact formulae for the complete k-partite graphs, split graphs, generalized convex graphs, and several planar grids, including rectangular grids, triangular grids, and triangulated-rectangular grids.

1-Embeddability of complete multipartite graphs on the projective plane

A graph is 1-embeddable on a closed surface if there exists a drawing of the graph on the surface such that each edge crosses at most one other edge at a point. In this paper, we determine all the 1-embeddable complete k-partite graphs on the projective plane.

Decompositions of Some Classes of Dense Graphs into Cycles of Lengths 4 and 8

For an even graph G and non-negative integers r and s, the pair $$(r,\,s)$$ is an admissible pair if $$4r+8s=|E(G)|.$$ If G admits a decomposition into r copies of $$C_4,$$ the cycle of length four, and s copies of $$C_8,$$ the cycle of length eight, for every admissible pair $$(r,\,s),$$ then G has a $$\{C_4^r,\,C_8^s\}$$ -decomposition. In this paper, a necessary and sufficient condition is obtained for the existence of a $$\{C_4^r,\,C_8^s\}$$ -decomposition of the complete k-partite graph $$K_{a_1,\,a_2,\,\dots ,\,a_k},$$ where $$k\ge 3.$$ Further, a characterization is obtained for the graph $$K_m\times K_n,$$ where $$\times$$ denotes the tensor product of graphs, to admit a $$\{C_4^r,\,C_8^s\}$$ -decomposition.

Strong rainbow disconnection in graphs

Let G be a nontrivial edge-colored connected graph. A rainbow edge-cut is an edge-cut R of G, and all edges of R have different colors in G. For two different vertices u and v of G, a u-v-edge-cut is an edge-cut separating them. An edge-colored graph G is called strong rainbow disconnected if for every two distinct vertices u and v of G, there exists a both rainbow and minimum u-v-edge-cut in G, and such an edge-coloring is called a strong rainbow disconnection coloring (srd-coloring for short) of G. For a connected graph G, the strong rainbow disconnection number (srd-number for short) of G, denoted by srd(G), is the minimum number of colors required to make G strong rainbow disconnected. In this paper, we first characterize the graphs with m edges satisfing srd(G) = k for each k ∈ {1, 2, m}, respectively, and we also show that the srd-number of a nontrivial connected graph G is equal to the maximum srd-number in the blocks of G. Secondly, we study the srd-numbers for the complete k-partite graphs, k-edge-connected k-regular graphs and grid graphs. Finally, we prove that for a connected graph G, computing srd(G) is NP-hard. In particular, we prove that it is NP-complete to decide if srd(G) = 3 for a connected cubic graph. We also show that the following problem is NP-complete: given an edge-colored (with an unbounded number of colors) connected graph G, check whether the given coloring makes G strong rainbow disconnected.

Proof of a conjecture on distance energy change of complete multipartite graph due to edge deletion

Let Kn1,n2,…,nk be a complete k-partite graph with k≥2 and ni≥2 for i=1,2,…,k. The Turán graph T(n,k) is a complete k-partite graph of n vertices with sizes of partitions as equal as possible. The distance energy ED(G) of a graph G is defined as the sum of absolute values of distance eigenvalues of the graph G. Varghese et al. (2018) [11] conjectured thatED(Kn1,n2,…,nk)<ED(Kn1,n2,…,nk−e), where e is any edge of Kn1,n2,…,nk and proved that the above relation holds for k=2. Very recently, Tian et al. (2020) [10] confirmed that the above conjecture holds for T(n,k) with n≡0(modk) and T(n,3). They also mentioned a weaker conjecture as follows:ED(T(n,k))<ED(T(n,k)−e), where e is any edge of T(n,k) and k≥2, n≥2k. In this paper, we confirm that the former conjecture is true for k≥3 and then the latter conjecture follows immediately.

Normalized Laplacian spectrum of complete multipartite graphs

The spectrum of the normalized Laplacian matrix of a graph provides many structural information of the graph, and it has many applications in numerous areas and in different guises. Let G be a complete k-partite graph with k≥3. In this paper, we give the necessary and sufficient condition for G which is determined by their normalized Laplacian spectrum. Moreover, we obtain a majorization theory of normalized Laplacian spectral radius of G, which enables us to find the maximal and minimal normalized Laplacian spectral radii among all complete k-partite graphs with fixed order, respectively.

Some reduction operations to pairwise compatibility graphs

A graph G=(V,E) with a vertex set V and an edge set E is called a pairwise compatibility graph (PCG, for short) if there are a tree T whose leaf set is V, a non-negative edge weight w in T, and two non-negative reals dmin≤dmax such that G has an edge uv∈E if and only if the sum of the edge weights in the unique path between u and v in the weighted tree (T,w) is in the interval [dmin,dmax]. PCG is a relatively new graph class motivated from bioinformatics. In this paper, we give some necessary conditions and sufficient conditions for a graph to be in PCG based on cut-vertices and twins, which provide reductions among PCGs. Our results imply that complete k-partite graphs, cacti, and some other graph classes are subsets of PCG.

Minimizing the number of 5-cycles in graphs with given edge-density

AbstractMotivated by the work of Razborov about the minimal density of triangles in graphs we study the minimal density of the 5-cycle C5. We show that every graph of order n and size $ (1 - 1/k) \left( {\matrix{n \cr 2 }} \right) $, where k ≥ 3 is an integer, contains at least $$({1 \over {10}} - {1 \over {2k}} + {1 \over {{k^2}}} - {1 \over {{k^3}}} + {2 \over {5{k^4}}}){n^5} + o({n^5})$$ copies of C5. This bound is optimal, since a matching upper bound is given by the balanced complete k-partite graph. The proof is based on the flag algebras framework. We also provide a stability result. An SDP solver is not necessary to verify our proofs.

2-Coloring number revisited

2-Coloring number is a parameter, which is often used in the literature to bound the game chromatic number and other related parameters. However, this parameter has not been precisely studied before.In this paper we aim to fill this gap. In particular we show that the approximation of the game chromatic number by the 2-coloring number can be very poor for many graphs. Additionally we prove that the 2-coloring number may grow quadratically as a function of the maximum degree of a graph, whereas the game chromatic number is always at most linear.Moreover, we establish the values of the 2-coloring number for several graph classes, such as complete k-partite graphs, cacti and thorny graphs, trees and subcubic graphs. It is shown that in all these cases one may compute the exact value in a polynomial time.

Hilbert series of binomial edge ideals

Let G be a finite simple graph on n vertices and JG denote the corresponding binomial edge ideal in the polynomial ring In this article, we compute the Hilbert series of binomial edge ideal of decomposable graphs in terms of Hilbert series of its indecomposable subgraphs. Also, we compute the Hilbert series of binomial edge ideal of join of two graphs and as a consequence we obtain the Hilbert series of complete k-partite graph, fan graph, multi-fan graph and wheel graph.