Complete Bipartite Graph Kn Research Articles

Bipartite Ramsey numbers of cycles

Let H1, …, Hk be bipartite graphs. The bipartite Ramsey number br(H1,…,Hk) is the minimum integer N such that any k-edge-coloring of the complete bipartite graph KN,N contains a monochromatic Hi in color i for some i∈{1,2,…,k}. The study of bipartite Ramsey number was initiated in the early 70s by Faudree and Schelp (1975), and they showed that br(P2n,P2m)=n+m−1, where Pn denotes a path with n vertices. Combining their result and the regularity lemma, one can obtain asymptotic value br(C2⌊α1n⌋,C2⌊α2n⌋)=(α1+α2+o(1))n for α1,α2>0, where Cn denotes a cycle with n vertices. The study of bipartite Ramsey numbers of cycles has also attracted lots of attention recently. For example, Bucić, Letzter and Sudakov (2019) showed that br(C2n,C2n,C2n)=(3+o(1))n and they remarked that it is interesting to determine the exact value of br(C2n,C2m). Zhang and Sun (2011), Zhang, Sun, and Wu (2013), and Gholami and Rowshan (2021) determined br(C4,C2n), br(C6,C2n), and br(C8,C2n) respectively. In this paper, we determine br(C2n,C2m) for all the remaining cases and show that br(C2n,C2m)=n+m−1 for all n>m≥5 and br(C2n,C2n)=2n if n≥5. It is somehow interesting that br(C2n,C2m) is the same as br(P2n,P2m) if n≠m, and br(C2n,C2n) is one more than br(P2n,P2n).

Complete bipartite graphs without small rainbow subgraphs

Motivated by bipartite Gallai–Ramsey type problems, we consider edge-colorings of complete bipartite graphs without rainbow tree and matching. Given two graphs G and H, and a positive integer k, define the bipartite Gallai–Ramsey numberbgrk(G:H) as the minimum number of vertices n such that n2≥k and for every N≥n, any coloring (using all k colors) of the complete bipartite graph KN,N contains a rainbow copy of G or a monochromatic copy of H. In this paper, we first describe the structures of a complete bipartite graph Kn,n without rainbow P4+ and 3K2, respectively, where P4+ is the graph consisting of a P4 with one extra edge incident with an interior vertex. Furthermore, we determine the exact values or upper and lower bounds on bgrk(G:H) when G is a 3-matching or a 4-path or P4+, and H is a bipartite graph.

Complete bipartite graphs without small rainbow stars

The k-edge-colored bipartite Gallai–Ramsey number bgrk(G:H) is defined as the minimum integer n such that n2≥k and for every N≥n, every edge-coloring (using all k colors) of complete bipartite graph KN,N contains a rainbow copy of G or a monochromatic copy of H. In this paper, we first study the structural theorem on the complete bipartite graph Kn,n with no rainbow copy of K1,3. Next, we utilize the results to prove the exact values of bgrk(P4:H), bgrk(P5:H), bgrk(K1,3:H), where H is a various union of cycles and paths and stars.

ABC energies and spectral radii of some graph operations

The present article presents some new results relating to Atomic Bond Connectivity energies and Spectral radii of generalized splitting and generalized shadow graphs constructed on the basis of some fundamental families of cycle graph Cn, complete graph Kn and complete bipartite graph Kn,n referred as base graphs. In fact we relate the energies and Spectral radii of splitting and shadow graphs with the energies and Spectral radii of original graphs.

Multicolor bipartite Ramsey numbers for quadrilaterals and stars

For bipartite graphs H1,…,Hμ, μ≥2, the μ-color bipartite Ramsey number, denoted by Rb(H1,…,Hμ), is the least positive integer N such that if we arbitrarily color the edges of a complete bipartite graph KN,N with μ colors, then it contains a monochromatic copy of Hi in color i for some i, 1≤i≤μ. Let C4 and K1,n be a quadrilateral and a star on n+1 vertices, respectively. In this paper, we show that the (μ+1)-color bipartite Ramsey number Rb(C4,…,C4,K1,n)≤n+⌈12μ2(4n+μ2+2μ−7)+4⌉+μ2+μ2−1. Moreover, using algebraic methods, we construct Ramsey graphs or near Ramsey graphs and determine infinitely many values of Rb(C4,…,C4,K1,n), which reach the upper bound if μ=1,2 and are at most ⌊μ2⌋ less than the upper bound if μ≥3.

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.

Higher matching complexes of complete graphs and complete bipartite graphs

For r≥1, the r-matching complex of a graph G, denoted Mr(G), is a simplicial complex whose faces are the subsets H⊆E(G) of the edge set of G such that the degree of any vertex in the induced subgraph G[H] is at most r. In this article, we give a closed form formula for the homotopy type of the (n−2)-matching complex of complete graph on n vertices. We also prove that the (n−1)-matching complex of complete bipartite graph Kn,n is homotopy equivalent to a sphere of dimension (n−1)2−1.

The planted matching problem: Phase transitions and exact results

We study the problem of recovering a planted matching in randomly weighted complete bipartite graphs Kn,n. For some unknown perfect matching M∗, the weight of an edge is drawn from one distribution P if e∈M∗ and another distribution Q if e∉M∗. Our goal is to infer M∗, exactly or approximately, from the edge weights. In this paper we take P=exp(λ) and Q=exp(1/n), in which case the maximum-likelihood estimator of M∗ is the minimum-weight matching Mmin. We obtain precise results on the overlap between M∗ and Mmin, that is, the fraction of edges they have in common. For λ≥4 we have almost perfect recovery, with overlap 1−o(1) with high probability. For λ<4 the expected overlap is an explicit function α(λ)<1: we compute it by generalizing Aldous’ celebrated proof of the ζ(2) conjecture for the unplanted model, using local weak convergence to relate Kn,n to a type of weighted infinite tree, and then deriving a system of differential equations from a message-passing algorithm on this tree.

Almost color-balanced perfect matchings in color-balanced complete graphs

For a graph G and a not necessarily proper k-edge coloring c:E(G)→{1,…,k}, let mi(G) be the number of edges of G of color i, and call G color-balanced if mi(G)=mj(G) for every two colors i and j. Several famous open problems relate to this notion; Ryser's conjecture on transversals in latin squares, for instance, is equivalent to the statement that every properly n-edge colored complete bipartite graph Kn,n has a color-balanced perfect matching. We contribute some results on the question posed by Kittipassorn and Sinsap (arXiv:2011.00862v1) whether every k-edge colored color-balanced complete graph K2kn has a color-balanced perfect matching M. For a perfect matching M of K2kn, a natural measure for the total deviation of M from being color-balanced is f(M)=∑i=1k|mi(M)−n|. While not every k-edge colored color-balanced complete graph K2kn has a color-balanced perfect matching M, that is, a perfect matching with f(M)=0, we prove the existence of a perfect matching M with f(M)=O(kknln⁡(k)) for general k and f(M)≤2 for k=3; the case k=2 has already been studied earlier. An attractive feature of the problem is that it naturally invites the combination of a combinatorial approach based on counting and local exchange arguments with probabilistic and geometric arguments.

Effective resistances and spanning trees in the complete bipartite graph plus a matching

In 2019, Ye and Yan computed the effective resistances in the nearly balanced complete bipartite graph Kn,n−pK2(p≤n). Then the result was extended to Km,n−pK2(p≤min{m,n}) very recently. In this paper, we obtain the effective resistances and the number of spanning trees in any complete bipartite graph plus a matching.

Bipartite Ramsey numbers of [formula omitted] in many colors

For any bipartite graph B, the bipartite Ramsey number brk(B) is defined to be the smallest integer N such that any edge-coloring of complete bipartite graph KN,N by k colors contains a monochromatic B. In this note, it is shown that brk(K2,s)∼(s−1)k2 for fixed s≥2 and brk(K3,3)∼k3 as k→∞.

Metric dimension and pattern avoidance in graphs

In this paper, we prove a number of results about pattern avoidance in graphs with bounded metric dimension or edge metric dimension. We show that the maximum possible number of edges in a graph of diameter D and edge metric dimension k is at most (⌊2D3⌋+1)k+k∑i=1⌈D3⌉(2i)k−1, sharpening the bound of k2+kDk−1+Dk from Zubrilina (2018). We also show that the maximum value of n for which some graph of metric dimension ≤k contains the complete graph Kn as a subgraph is n=2k. We prove that the maximum value of n for which some graph of metric dimension ≤k contains the complete bipartite graph Kn,n as a subgraph is 2Θ(k). Furthermore, we show that the maximum value of n for which some graph of edge metric dimension ≤k contains K1,n as a subgraph is n=2k. We also show that the maximum value of n for which some graph of metric dimension ≤k contains K1,n as a subgraph is 3k−O(k).In addition, we prove that the d-dimensional grids ∏i=1dPri have edge metric dimension at most d. This generalizes two results of Kelenc et al. (2016), that non-path grids have edge metric dimension 2 and that d-dimensional hypercubes have edge metric dimension at most d. We also provide a characterization of n-vertex graphs with edge metric dimension n−2, answering a question of Zubrilina. As a result of this characterization, we prove that any connected n-vertex graph G such that edim(G)=n−2 has diameter at most 5. More generally, we prove that any connected n-vertex graph with edge metric dimension n−k has diameter at most 3k−1.

The hat guessing number of graphs

Consider the following hat guessing game: n players are placed on n vertices of a graph, each wearing a hat whose color is arbitrarily chosen from a set of q possible colors. Each player can see the hat colors of his neighbors, but not his own hat color. All of the players are asked to guess their own hat colors simultaneously, according to a predetermined guessing strategy and the hat colors they see, where no communication between them is allowed. Given a graph G, its hat guessing number HG(G) is the largest integer q such that there exists a guessing strategy guaranteeing at least one correct guess for any hat assignment of q possible colors.In 2008, Butler et al. asked whether the hat guessing number of the complete bipartite graph Kn,n is at least some fixed positive (fractional) power of n. We answer this question affirmatively, showing that for sufficiently large n, the complete r-partite graph Kn,…,n satisfies HG(Kn,…,n)=Ω(nr−1r−o(1)). Our guessing strategy is based on a probabilistic construction and other combinatorial ideas, and can be extended to show that HG(C→n,…,n)=Ω(n1r−o(1)), where C→n,…,n is the blow-up of a directed r-cycle, and where for directed graphs each player sees only the hat colors of his outneighbors.Additionally, we consider related problems like the relation between the hat guessing number and other graph parameters, and the linear hat guessing number, where the players are only allowed to use affine linear guessing strategies. Several nonexistence results are obtained by using well-known combinatorial tools, including the Lovász Local Lemma and the Combinatorial Nullstellensatz. Among other results, it is shown that under certain conditions, the linear hat guessing number of Kn,n is at most 3, exhibiting a huge gap from the Ω(n12−o(1)) (nonlinear) hat guessing number of this graph.

A 4-choosable graph that is not (8:2)-choosable

List coloring is a generalization of graph coloring introduced by Erdős, Rubin and Taylor in 1980, which has become extensively studied in graph theory. A graph G is said to be k-choosable, or k-list-colorable, if, for every way of assigning a list (set) of k colors to each vertex of G, it is possible to choose a color from each list in such a way that no two neighboring vertices receive the same color. Note that if the lists are all the same, then this is asking for G to have chromatic number at most k. One might think that the case where all the lists are the same would be the hardest: surely making the lists different should make it easier to ensure that neighboring vertices have different colors. Rather surprisingly, however, this is not the case. A counterexample is provided by the complete bipartite graph K2,4. If the two vertices in the first vertex class are assigned the lists {a,b} and {c,d}, while the vertices in the other vertex class are assigned the lists {a,c}, {a,d}, {b,c} and {b,d}, then it is easy to check that it is not possible to obtain a proper coloring from these lists, so G is not 2-choosable, and yet the chromatic number of G is 2. A famous theorem of Galvin, which solved the so-called Dinitz conjecture, states that the line graph of the complete bipartite graph Kn,n is n-choosable. Equivalently, if each square of an n×n grid is assigned a list of n colors, it is possible to choose a color from each list in such a way that no color appears more than once in any row or column. One can generalize this notion by requiring a choice of not just one color from each list, but some larger number of colors. A graph G is said to be (A,B)-list-colorable if, for every assignment of lists to the vertices of G, each consisting of A colors, there is an assignment of sets of B colors to the vertices such that each vertex is assigned a set that is a subset of its list and the sets assigned to pairs of adjacent vertices are disjoint. (When B=1 this simply says that G is A-choosable.) In this short paper, the authors answer a question that has remained open for almost four decades since it was posed by Erdős, Rubin and Taylor in their seminal paper: if a graph is (A,B)-list-colorable, is it true that it is also (mA,mB)-list-colorable for every m≥1? Quite surprisingly, the answer is again negative - the authors construct a graph that is (4,1)-list-colorable but not (8,2)-list-colorable.

Graphs admitting antimagic labeling for arbitrary sets of positive numbers

Hartsfield and Ringel in 1990 conjectured that any connected graph with q≥2 edges has an edge labeling f with labels in the set {1,…,q}, such that for every two distinct vertices u and v, fu≠fv, where fv=∑e∈E(v)f(e), and E(v) is the set of edges of the graph incident to vertex v.We say that a graph G=(V,E), with q edges, is universal antimagic, if for every set B of q positive numbers there is a bijection f:E→B such that fu≠fv, for any two distinct vertices u and v. It is weighted universal antimagic if for any vertex weight function w and every set B of q positive numbers there is a bijection f:E→B such that w(u)+fu≠w(v)+fv, for any two distinct vertices u and v.In this work we prove that paths, cycles, and graphs whose connected components are cycles or paths of odd lengths are universal antimagic. We also prove that a split graph and any graph containing a complete bipartite graph as a spanning subgraph is universal antimagic. Surprisingly, we are also able to prove that any graph containing a complete bipartite graph Kn,m with n,m≥3 as a spanning subgraph is weighted universal antimagic. From all the results we can derive effective methods to construct the labelings.

Complete graphs and complete bipartite graphs without rainbow path

Motivated by Ramsey-type questions, we consider edge-colorings of complete graphs and complete bipartite graphs without rainbow path. Given two graphs G and H, the k-colored Gallai–Ramsey number grk(G:H) is defined to be the minimum integer n such that n2≥k and for every N≥n, every rainbow G-free coloring (using all k colors) of the complete graph KN contains a monochromatic copy of H. In this paper, we first provide some exact values and bounds of grk(P5:Kt). Moreover, we define the k-colored bipartite Gallai–Ramsey number bgrk(G:H) as the minimum integer n such that n2≥k and for every N≥n, every rainbow G-free coloring (using all k colors) of the complete bipartite graph KN,N contains a monochromatic copy of H. Furthermore, we describe the structures of complete bipartite graph Kn,n with no rainbow P4 and P5, respectively. Finally, we find the exact values of bgrk(P4:Ks,t) (1≤s≤t), bgrk(P4:F) (where F is a subgraph of Ks,t), bgrk(P5:K1,t) and bgrk(P5:K2,t) by using the structural results.

On even-closed regular embeddings of graphs

A map is said to be even-closed if all of its automorphisms act like even permutations on the vertex set. In this paper the study of even-closed regular maps is approached by analysing two distinguished families. The first family consists of embeddings of a well-known family of graphs on distinct orientable surfaces, whereas in the second family we consider all graphs having orientable-regular embeddings on a particular surface. In particular, the classification of even-closed orientable-regular embeddings of the complete bipartite graphs Kn,n and classification of even-closed orientable-regular maps on the torus are given.

On the representation number of a crown graph

A graph G=(V,E) is word-representable if there exists a word w over the alphabet V such that letters x and y alternate in w if and only if xy is an edge in E. It is known (Kitaev and Pyatkin, 2008) that any word-representable graph G is k-word-representable for some k, that is, there exists a word w representing G such that each letter occurs exactly k times in w. The minimum such k is called G’s representation number.A crown graph (also known as a cocktail party graph) Hn,n is a graph obtained from the complete bipartite graph Kn,n by removing a perfect matching. In this paper, we show that for n≥5, Hn,n’s representation number is ⌈n∕2⌉. This result not only provides a complete solution to the open Problem 7.4.2 in Kitaev and Lozin (2015), but also gives a negative answer to the question raised in Problem 7.2.7 in Kitaev and Lozin (2015) on 3-word-representability of bipartite graphs. As a byproduct, we obtain a new example of a graph class with a high representation number.

Robust cycle bases do not exist for [formula omitted] if [formula omitted

A basis for the cycle space of a graph is said to be robust if any cycle Z of G is a sum Z=C1+C2+⋯+Ck of basis elements such that (i) (C1+C2+⋯+Cℓ−1)∩Cℓ is a nontrivial path for each 2≤ℓ<k. Hence, (ii) each partial sum C1+C2+⋯+Cℓ is a cycle for 1≤ℓ≤k. While complete graphs and 2-connected plane graphs have robust cycle bases, it is shown that regular complete bipartite graphs Kn,n do not have any robust cycle basis if n≥8.

On sequences of polynomials arising from graph invariants

Graph polynomials are deemed useful if they give rise to algebraic characterizations of various graph properties, and their evaluations encode many other graph invariants. Algebraic: The complete graphs Kn and the complete bipartite graphs Kn,n can be characterized as those graphs whose matching polynomials satisfy a certain recurrence relations and are related to the Hermite and Laguerre polynomials. An encoded graph invariant: The absolute value of the chromatic polynomial χ(G,X) of a graph G evaluated at −1 counts the number of acyclic orientations of G.In this paper we prove a general theorem on graph families which are characterized by families of polynomials satisfying linear recurrence relations. This gives infinitely many instances similar to the characterization of Kn,n. We also show where to use, instead of the Hermite and Laguerre polynomials, linear recurrence relations where the coefficients do not depend on n.Finally, we discuss the distinctive power of graph polynomials in specific form.