Complete Graph Kn Research Articles

Proof of a conjecture on communicability distance sum index of graphs

Let G be a connected graph with adjacency matrix A, and let A=exp⁡(A). The communicability distance between two vertices u and v of G is defined as ξuv=(Auu+Avv−2Auv)1/2, and the communicability distance sum index (CDS index for short) of G is the sum of all communicability distances between vertices of G. In this paper, it is shown that the complete graph Kn is the unique graph attaining the minimum CDS index among all connected graphs of order n. This confirms a conjecture of Estrada (2012) [2]. Furthermore, some upper and lower bounds for the CDS index of graphs are provided.

Enumeration of extensions of the cycle matroid of a complete graph

We prove that the number of single element extensions of the cycle matroid of the complete graph Kn+1 is 2(nn/2)(1+o(1)). This is done using a characterization of extensions as “linear subclasses”.

Some signed graphs whose eigenvalues are main

Let G be a graph. For a subset X of V(G), the switching σ of G is the signed graph Gσ obtained from G by reversing the signs of all edges between X and V(G)∖X. Let A(Gσ) be the adjacency matrix of Gσ. An eigenvalue of A(Gσ) is called a main eigenvalue if it has an eigenvector the sum of whose entries is not equal to zero. Let Sn,k be the graph obtained from the complete graph Kn−r by attaching r pendent edges at some vertex of Kn−r. In this paper we prove that there exists a switching σ such that all eigenvalues of Gσ are main when G is a complete multipartite graph, or G is a harmonic tree, or G is a Sn,k. These results partly confirm a conjecture of Akbari et al.

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.

On the unimodality of average edge cover polynomials

The edge cover polynomial of a graph G is the function E(G,x)=∑i≥1e(G,i)xi, where e(G,i) is the number of edge coverings of G with size i. In this paper, we show that the average edge cover polynomial of order n is reduced to the edge cover polynomial of complete graph Kn, based on which we establish that the average edge cover polynomial of order n is unimodal and has at least n−3 non-real roots.

On color isomorphic subdivisions

Given a graph H and an integer k⩾2, let fk(n,H) be the smallest number of colors C such that there exists a proper edge-coloring of the complete graph Kn with C colors containing no k vertex-disjoint color isomorphic copies of H. In this paper, we prove that f2(n,Ht)=Ω(n1+12t−3) where Ht is the 1-subdivision of the complete graph Kt. This answers a question of Conlon and Tyomkyn (2021) [4].

Vertex pancyclicity over lexicographic products

A lexicographic product of graphs G and H, denoted by is defined as a graph with the vertex set and an edge presents in the product whenever or (u 1 = u 2 and ). We investigate the sufficient conditions for vertex pancyclicity of lexicographic products of complete graphs Kn , paths Pn or cycles Cn with a general graph. We obtain that (i) if G 1 is a traceable graph of even order and G 2 is a graph with at least one edge, then is vertex pancyclic; (ii) if G 1 is hamiltonian and G 2 is a graph with at least one edge, then is vertex pancyclic.

The sharp K4-percolation threshold on the Erdős–Rényi random graph

We locate the critical threshold pc∼1∕3nlogn at which it becomes likely that the complete graph Kn can be obtained from the Erdős–Rényi graph Gn,p by iteratively completing copies of K4 minus an edge. This refines work of Balogh, Bollobás and Morris that bounds the threshold up to multiplicative constants.

Rainbow Perfect and Near-Perfect Matchings in Complete Graphs with Edges Colored by Circular Distance

Given an edge-colored complete graph Kn on n vertices, a perfect (respectively, near-perfect) matching M in Kn with an even (respectively, odd) number of vertices is rainbow if all edges have distinct colors. In this paper, we consider an edge coloring of Kn by circular distance, and we denote the resulting complete graph by K●n. We show that when K●n has an even number of vertices, it contains a rainbow perfect matching if and only if n=8k or n=8k+2, where k is a nonnegative integer. In the case of an odd number of vertices, Kirkman matching is known to be a rainbow near-perfect matching in K●n. However, real-world applications sometimes require multiple rainbow near-perfect matchings. We propose a method for using a recursive algorithm to generate multiple rainbow near-perfect matchings in K●n.

Topological Indices of Families of Bistar and Corona Product of Graphs

Topological indices are graph invariants that are used to correlate the physicochemical properties of a chemical compound with its (molecular) graph. In this study, we study certain degree‐based topological indices such as Randić index, Zagreb indices, multiplicative Zagreb indices, Narumi–Katayama index, atom‐bond connectivity index, augmented Zagreb index, geometric‐arithmetic index, harmonic index, and sum‐connectivity index for the bistar graphs and the corona product , where represents the complement of complete graph Kn.

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.

Gallai-Ramsey number of even cycles with chords

For a graph H and an integer k≥1, the k-color Ramsey number Rk(H) is the least integer N such that every k-coloring of the edges of the complete graph KN contains a monochromatic copy of H. Let Cm denote the cycle on m≥4 vertices and let Θm denote the family of graphs obtained from Cm by adding an additional edge joining two non-consecutive vertices. Unlike Ramsey number of odd cycles, little is known about the general behavior of Rk(C2n) except that Rk(C2n)≥(n−1)k+n+k−1 for all k≥2 and n≥2. In this paper, we study Ramsey number of even cycles with chords under Gallai colorings, where a Gallai coloring is a coloring of the edges of a complete graph without rainbow triangles. For an integer k≥1, the Gallai-Ramsey number GRk(H) of a graph H is the least positive integer N such that every Gallai k-coloring of the complete graph KN contains a monochromatic copy of H. We prove that GRk(Θ2n)=(n−1)k+n+1 for all k≥2 and n≥3. This implies that GRk(C2n)=(n−1)k+n+1 all k≥2 and n≥3. Our result yields a unified proof for the Gallai-Ramsey number of all even cycles on at least four vertices.

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.

Construction of L-Borderenergetic Graphs

If a graph G of order n has the Laplacian energy same as that of complete graph Kn then G is said to be L-borderenergeic graph. It is interesting and challenging as well to identify the graphs which are L-borderenergetic as only few graphs are known to be L-borderenergetic. In the present work we have investigated a sequence of L-borderenergetic graphs and also devise a procedure to find sequence of L-borderenergetic graphs from the known L-borderenergetic graph.

Bounded VC-Dimension Implies the Schur-Erdős Conjecture

In 1916, Schur introduced the Ramsey number r(3; m), which is the minimum integer n > 1 such that for any m-coloring of the edges of the complete graph Kn, there is a monochromatic copy of K3. He showed that r(3; m) ≤ O(m!), and a simple construction demonstrates that r(3; m) ≥ 2Ω(m). An old conjecture of Erdős states that r(3; m) = 2Θ(m). In this note, we prove the conjecture for m-colorings with bounded VC-dimension, that is, for m-colorings with the property that the set system induced by the neighborhoods of the vertices with respect to each color class has bounded VC-dimension.

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.

The threshold bias of the clique-factor game

Let r≥4 be an integer and consider the following game on the complete graph Kn for n∈rZ: Two players, Maker and Breaker, alternately claim previously unclaimed edges of Kn such that in each turn Maker claims one and Breaker claims b∈N edges. Maker wins if her graph contains a Kr-factor, that is a collection of n/r vertex-disjoint copies of Kr, and Breaker wins otherwise. In other words, we consider a b-biased Kr-factor Maker–Breaker game. We show that the threshold bias for this game is of order n2/(r+2). This makes a step towards determining the threshold bias for making bounded-degree spanning graphs and extends a result of Allen et al. who resolved the case r∈{3,4} up to a logarithmic factor.

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.

Sharp inequalities for maximal operators on finite graphs, II

Let MG be the centered Hardy-Littlewood maximal operator on a finite graph G. We find limp→∞‖MG‖pp when G is the start graph (Sn) and the complete graph (Kn), and we fully describe ‖MSn‖p and the corresponding extremizers for p∈(1,2). We prove that limp→∞‖MSn‖pp=1+n2 when n≥25. Also, we compute the best constant CSn,2 such that for every f:V→R we have Var2MSnf≤CSn,2Var2f. We prove that CSn,2=(n2−n−1)1/2n for all n≥3 and characterize the extremizers. Moreover, when M is the Hardy-Littlewood maximal operator on Z, we compute the best constant Cp such that VarpMf≤Cp‖f‖p for p∈(12,1) and we describe the extremizers.

Gallai–Ramsey numbers for graphs with chromatic number three

Given a graph H and an integer k≥1, the Gallai–Ramsey number GRk(H) is defined to be the minimum integer n such that every k-edge coloring of the complete graph Kn contains either a rainbow (all different colored) triangle or a monochromatic copy of H. In this paper, we study Gallai–Ramsey numbers for graphs with chromatic number three such as K̂m for m≥2, where K̂m is a kipas with m+1 vertices obtained from the join of K1 and Pm, and a class of graphs with five vertices, denoted by ℋ. We first study the general lower bound of such graphs and propose a conjecture for the exact value of GRk(K̂m). Then we give a unified proof to determine the Gallai–Ramsey numbers for many graphs in ℋ and obtain the exact value of GRk(K̂4) for k≥1. Our outcomes not only indicate that the conjecture on GRk(K̂m) is true for m≤4, but also imply several results on GRk(H) for some H∈ℋ which are proved individually in different papers.