Complete Graph Kn Research Articles

Extremal paths in inhomogenous random graphs

In this paper, we study long open paths in an inhomogenous Erdős–Rényi random graph G obtained from the complete graph Kn on n vertices by allowing each edge e to be open with probability pn(e), independently of other edges. If the edge probability assignment sequence satisfies certain neighbour density conditions, then G has a long path containing nearly all the vertices with high probability, i.e., with probability converging to one as n→∞. Our methods extend to random weighted graphs with nonidentical weight distributions and we describe conditions under which the minimum weight Hamiltonian path has weight bounded above by a constant, with high probability.

The genus of complete 3-uniform hypergraphs

In 1968, Ringel and Youngs confirmed the last open case of the Heawood Conjecture by determining the genus of every complete graph Kn. In this paper, we investigate the minimum genus embeddings of the complete 3-uniform hypergraphs Kn3. Embeddings of a hypergraph H are defined as the embeddings of its associated Levi graph LH with vertex set V(H)⊔E(H), in which v∈V(H) and e∈E(H) are adjacent if and only if v and e are incident in H. We determine both the orientable and the non-orientable genus of Kn3 when n is even. Moreover, it is shown that the number of non-isomorphic minimum genus embeddings of Kn3 is at least 214n2log⁡n(1−o(1)). The construction in the proof may be of independent interest as a design-type problem.

On circular–linear one-factorizations of the complete graph

In Korchmáros et al. (2018)one-factorizations of the complete graph Kn are constructed for n=q+1 with any odd prime power q such that either q≡1(mod4) or q=2h−1. The arithmetic restriction n=q+1 is due to the fact that the vertices of Kn in the construction are the points of a conic Ω in the finite plane of order q. Here we work on the Euclidean plane and describe an analogous construction where the role of Ω is taken by a regular n-gon. This allows us to remove the above constraints and construct one-factorizations of Kn for every even n≥6.

Packing degenerate graphs

Given D and γ>0, whenever c>0 is sufficiently small and n sufficiently large, if G is a family of D-degenerate graphs of individual orders at most n, maximum degrees at most cnlog⁡n, and total number of edges at most (1−γ)(n2), then G packs into the complete graph Kn. Our proof proceeds by analysing a natural random greedy packing algorithm.

On Edge Irregular Total k-labeling and Total Edge Irregularity Strength of Barbell Graphs

Let G be a connected graph with a non empty vertex set V(G) and edge set E(G). An edge irregular total k-labeling of a graph G is a labeling λ : V(G) ⋃ E(G) → {1, 2, …, k}, so that every two different edges have different weights. The weight of edge uv of G is the sum of the labels vertices u and v and label of the edge uv, which is can be written as wt(uv) = λ (u) + λ (uv) + λ(v). The total edge irregularity strength of G, denoted by tes(G) is the minimum positive integer k for which the graph G has an edge irregular total k-labeling. Barbell graph Bn is obtained by connecting two copies of a complete graph Kn by a bridge. In this research, we determined the total edge irregularity strength of barbell graph Bn for n ≥ 3.

Laplacian borderenergetic graphs

The graph G of order n is said to be Laplacian (signless Laplacian) borderenergetic if its Laplacian (signless Laplacian) energy equals the Laplacian (signless Laplacian) energy of the complete graph Kn. In this paper, we determine the Laplacian (signless Laplacian) borderenergetic graphs of order n, where n ≤ 9.

Density of Monochromatic Infinite Subgraphs

For any countably infinite graph G, Ramsey’s theorem guarantees an infinite monochromatic copy of G in any r-coloring of the edges of the countably infinite complete graph Kℕ. Taking this a step further, it is natural to wonder how “large” of a monochromatic copy of G we can find with respect to some measure - for instance, the density (or upper density) of the vertex set of G in the positive integers. Unlike finite Ramsey theory, where this question has been studied extensively, the analogous problem for infinite graphs has been mostly overlooked. In one of the few results in the area, Erdős and Galvin proved that in every 2-coloring of Kℕ, there exists a monochromatic path whose vertex set has upper density at least 2/3, but it is not possible to do better than 8/9. They also showed that for some sequence en → 0, there exists a monochromatic path P such that for infinitely many n, the set (1,2,...,n contains the first $$\left(\frac{1}{3+\sqrt{3}}-\epsilon_n\right)n$$ vertices of P, but it is not possible to do better than 2n/3. We improve both results, in the former case achieving an upper density at least 3/4 and in the latter case obtaining a tight bound of 2/3. We also consider related problems for directed paths, trees (connected subgraphs), and a more general result which includes locally finite graphs for instance.

A new lower bound for the energy of graphs

Let G be a simple graph with n vertices v1,…,vn and m edges. Let A(G)=[aij]n×n be the adjacency matrix of G, that is aij=1 if vi and vj are adjacent, and aij=0, otherwise. Let λ1,…,λn be the eigenvalues of A(G). The energy of G, denoted by E(G), is defined as |λ1|+⋯+|λn|. It is well known that 2m≤E(G)≤2mn. In this paper we improve the latter lower bound and prove that if all eigenvalues of G are non-zero, thenE(G)≥2|λ1λn||λ1|+|λn|2mn, where |λ1|≥⋯≥|λn|. We show that the equality holds if and only if G=n2K2 or G=pKβ+1⋃qTβ+1, where p and q are some non-negative integers and β≥2 and Tβ+1 is the graph Kβ+1,β+1∖M, where M is a perfect matching of Kβ+1,β+1. Finally by studying the |λ1λn||λ1|+|λn| of graphs, we find some lower bounds for the energy of graphs. In particular, we obtain that if G has no eigenvalue in the interval (−1,1), thenE(G)≥2m2n−2n, and the equality holds if and only if G is the complete graph Kn.

The rainbow connectivity of cartesian product graphs

An edge-coloured graph G is said to be rainbow-connected if any two vertices are connected by a path whose edges have different colours. The rainbow connection number of a graph is the minimum number of colours needed to make the graph rainbow-connected. This parameter was introduced by G. Chartrand, G.L. Johns, K.A. McKeon and P. Zhang in 2008. Similar to rainbow connection coloring, an edge-coloring is a rainbow k-connection coloring if there are at least k internally disjoint rainbow u – v paths connecting any two distinct vertices u and v. And the rainbow k-connectivity rck (G) of G to be the minimum integer l such that there exists a l-edge-coloring which is a rainbow k-connection coloring. In [4], the authors determined the rck (G) of the complete graph Kn and r-regular complete bipartite graphs Kr, r . In this paper, we study the rainbow k-connectivity of the Cartesian product graphs K 2□Kn for some k. We determine rck (K 2□Kn ) for k = 2, 3, 4 and prove that there exists Cartesian product graphs K 2□Kn such that rck (K 2□Kn ) = 3 for each integer k ≥ 2.

Monochromatic subgraphs in randomly colored graphons

Let T(H,Gn) be the number of monochromatic copies of a fixed connected graph H in a uniformly random coloring of the vertices of the graph Gn. In this paper we give a complete characterization of the limiting distribution of T(H,Gn), when {Gn}n≥1 is a converging sequence of dense graphs. When the number of colors grows to infinity, depending on whether the expected value remains bounded, T(H,Gn) either converges to a finite linear combination of independent Poisson variables or a normal distribution. On the other hand, when the number of colors is fixed, T(H,Gn) converges to a (possibly infinite) linear combination of independent centered chi-squared random variables. This generalizes the classical birthday problem, which involves understanding the asymptotics of T(Ks,Kn), the number of monochromatic s-cliques in a complete graph Kn (s-matching birthdays among a group of n friends), to general monochromatic subgraphs in a network.

A [formula omitted]-decomposition of the [formula omitted]-fold line graph of the complete graph

In 1996, Cox and Rodger [Cycle systems of the line graph of the complete graph, J. Graph Theory 21 (1996) 173–182] raised the following question: For what values of m and n does there exist an m-cycle decomposition of L(Kn)? In this paper, the above question is answered for m=5. In fact, it is shown that L(Kn)(λ), the λ-fold line graph of the complete graph Kn, has a C5-decomposition if and only if 5∣λn2(n−2) and n≥4.

Counting Gallai 3-colorings of complete graphs

An edge coloring of the n-vertex complete graph Kn is a Gallai coloring if it does not contain any rainbow triangle, that is, a triangle whose edges are colored with three distinct colors. We prove that the number of Gallai colorings of Kn with at most three colors is at most 7(n+1)2n2, which improves the best known upper bound of 32(n−1)!⋅2n−12 in Benevides et al. (2017) .

Sharp upper bounds of the spectral radius of a graph

Let G be a simple connected graph with n vertices and m edges. The spectral radiusρ(G) of G is the largest eigenvalue of its adjacency matrix. In this paper, we firstly consider the effect on the spectral radius of a graph by removing a vertex, and then as an application of the result, we obtain a new sharp upper bound of ρ(G) which improves some known bounds: If (k−2)(k−3)2≤m−n≤k(k−3)2, where k(3≤k≤n) is an integer, then ρ(G)≤2m−n−k+52+2m−2n+94.The equality holds if and only if G is a complete graph Kn or K4−e, where K4−e is the graph obtained from K4 by deleting some edge e.

Total rainbow connection numbers of some special graphs

In 2008, Chartrand et al. first introduced the concept of rainbow connection. Since then the study of rainbow connection has received considerable attention in the literature, and now it becomes an active topic in graph theory. As a natural generalization, Uchizawa et al. (2013) and Liu et al. (2014) presented the concept of total rainbow connection, respectively. In this paper, we investigate the total rainbow connection numbers of outerplanar graphs with diameter 2. Applying our result, we improve the main result of [X. Huang, X. Li, Y. Shi, J. Yue, Y. Zhao, Rainbow connections for outerplanar graphs with diameter 2 and 3, Applied Mathematics and Computation, 242(2014), 277–280]. Next, we revise the main result of [Y. Liu, Z. Wang, Rainbow Connection Number of the Thorn Graph, Applied Mathematical Sciences, 8(2014), 6373–6377], and determine the total rainbow connection numbers of graphs G, where G are the thorn graph of complete graph Kn*, the thorn graph of the cycle Cn*. At last, we study the rainbow 2-connection numbers of some special graphs.

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.

Hamiltonian Maker–Breaker Games on Small Graphs

We look at the unbiased Maker–Breaker Hamiltonicity game played on the edge set of a complete graph Kn , where Maker’s goal is to claim a Hamiltonian cycle. First, we prove that, independent of who starts, Maker can win the game for n = 8 and n = 9. Then we use an inductive argument to show that, independent of who starts, Maker can win the game if and only if This, in particular, resolves in the affirmative the long-standing conjecture of Papaioannou. We also study two standard positional games related to Hamiltonicity game. For Hamiltonian Path game, we show that Maker can claim a Hamiltonian path if and only if independent of who starts. Next, we look at Fixed Hamiltonian Path game, where the goal of Maker is to claim a Hamiltonian path between two predetermined vertices. We prove that if Maker starts the game, he wins if and only if and if Breaker starts, Maker wins if and only if Using this result, we are able to improve the previously best upper bound on the smallest number of edges a graph on n vertices can have, knowing that Maker can win the Maker–Breaker Hamiltonicity game played on its edges. To resolve the outcomes of the mentioned games on small (finite) boards, we devise algorithms for efficiently searching game trees and then obtain our results with the help of a computer.

On the total vertex irregularity strength of Cn *2 Kn graph

Let G be a connected and undirected graph with vertex set V (G) and edge set E(G). A vertex irregular total k-labeling of a graph is an assignment of positive integers set {1, 2, 3, … k} so that the weight for all vertices are distinct. The weight of vertex u in G, denoted by wt(u), is defined as the sum of the label of vertex u and the label of all edges incident with vertex u. Total vertex irregularity strength of G, denoted by tvs(G), is the minimum value of the largest label k over all such vertex irregular total k-labelings. The Cn *2 Kn graph is a graph obtained by edge operation amalgamation of cycle graph Cn with complete graph Kn. In this paper, we investigate the total vertex irregularity strength of Cn *2 Kn graphs, for n ≥ 3. The result is for n ≥ 3, 4 and for n ≥ 5

Quadrangular embeddings of complete graphs and the Even Map Color Theorem

Hartsfield and Ringel constructed orientable quadrangular embeddings of the complete graph Kn for n≡5(mod8), and nonorientable ones for n≥9 and n≡1(mod4). These provide minimal quadrangulations of their underlying surfaces. We extend these results to determine, for every complete graph Kn, n≥4, the minimum genus, both orientable and nonorientable, for the surface in which Kn has an embedding with all faces of degree at least 4, and also for the surface in which Kn has an embedding with all faces of even degree. These last embeddings provide sharpness examples for a result of Hutchinson bounding the chromatic number of graphs embedded with all faces of even degree, completing the proof of the Even Map Color Theorem. We also show that if a connected simple graph G has a perfect matching and a cycle then the lexicographic product G[K4] has orientable and nonorientable quadrangular embeddings; this provides new examples of minimal quadrangulations.

On Multiplicity of Quadrilaterals in Complete Graphs

In this paper, we determine a lower bound for the multiplicity of quadrilaterals (cycles on four vertices) in complete graph Kn for any positive integer n.

Minimum degree and size conditions for the proper connection number of graphs

An edge-coloured graph G is called properly connected if every two vertices are connected by a proper path. The proper connection number of a connected graph G, denoted by pc(G), is the smallest number of colours that are needed in order to make G properly connected. van Aardt et al. (2017)gave a sufficient condition for the proper connection number to be at most k in terms of the size of graphs. In this note, our main result is the following, by adding a minimum degree condition: let G be a connected graph of order n, k ≥ 3. If |E(G)|≥(n−m−(k+1−m)(δ+1)2)+(k+1−m)(δ+12)+k+2, then pc(G) ≤ k, where m takes the value k+1 if δ=1 and ⌊kδ−1⌋ if δ ≥ 2. Furthermore, if k=2 and δ=2,pc(G) ≤ 2, except G ∈ {G1, Gn} (n ≥ 8), where G1=K1∨3K2 and Gn is obtained by taking a complete graph Kn−5 and K1∨(2K2) with an arbitrary vertex of Kn−5 and a vertex with d(v)=4 in K1∨(2K2) being joined. If k=2,δ ≥ 3, we conjecture pc(G) ≤ 2, where m takes the value 1 if δ=3 and 0 if δ ≥ 4 in the assumption.