Graph Kn Research Articles

Gallai–Ramsey numbers for multiple triangles

A Gallai coloring of a complete graph is an edge-coloring such that no triangle has all its edges colored differently. A Gallai k-coloring is a Gallai coloring that uses at most k colors. Given an integer k≥2, and graphs H1,H2,…,Hk, the Gallai-Ramsey number GR(H1,H2,…,Hk) is the least positive integer N such that every Gallai k-coloring of the complete graph KN contains a monochromatic copy of Hi in color i for some i∈{1,2,…,k}. Let mG denote the union of m disjoint copies of G. This paper studies the Gallai–Ramsey number GR(t1K3,…,tkK3) for all integers t1≥t2≥⋯≥tk≥1. The case t1=t2=⋯=tk=1 was obtained by Chung and Graham (1983). In this paper, we obtain a general bound of GR(t1K3,…,tkK3) for all t1≥t2≥⋯≥tk≥1, and prove that both the upper and lower bounds can be attained for some ti, i∈{1,…,k}.

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→∞.

An Improvement of the Lower Bound on the Minimum Number of ≤k-Edges

In this paper, we improve the lower bound on the minimum number of ≤k-edges in sets of n points in general position in the plane when k is close to n2. As a consequence, we improve the current best lower bound of the rectilinear crossing number of the complete graph Kn for some values of n.

On the Second-Largest Reciprocal Distance Signless Laplacian Eigenvalue

The signless Laplacian reciprocal distance matrix for a simple connected graph G is defined as RQ(G)=diag(RH(G))+RD(G). Here, RD(G) is the Harary matrix (also called reciprocal distance matrix) while diag(RH(G)) represents the diagonal matrix of the total reciprocal distance vertices. In the present work, some upper and lower bounds for the second-largest eigenvalue of the signless Laplacian reciprocal distance matrix of graphs in terms of various graph parameters are investigated. Besides, all graphs attaining these new bounds are characterized. Additionally, it is inferred that among all connected graphs with n vertices, the complete graph Kn and the graph Kn−e obtained from Kn by deleting an edge e have the maximum second-largest signless Laplacian reciprocal distance eigenvalue.

The Domination number of 2-neighbourhood-corona graphs

Given a simple graph G. Vertex set of G is V, and edge set of G is E. Domination set, denoted by S, that is subset of V such that every vertex in V which is not element of S has distance one to S. The least number of the elements of S is the domination number of the graph G, that is ϒ(G). Let G1 and G2 be a simple graph. G1 has n1 vertices, and has m1 edges. G2 has n2 vertices, and has m2 edges. We defined an operator called neighbourhood corona, denoted by a star ‘*’. Graph G1*G2 is a new graph obtained by making ni copies of second graph and for each i make connecting all vertices in i-th copy of second graph G2 to neighbours of vi = 1, 2, …, n. Furthermore, new graph 2-neighbourhood corona G1*2G2, has n1 copies of G2 and for each i make connecting to all vertices of ith copy of G2 to neighbours of vi, i = 1, 2, …, n. In this research, we determined ϒ(Gi*2G2) where G1 is a complete graph Kn or Cycle Cn, and G2 is K1 or P2. Furthermore, we determined ϒ(G1 *mG2) due to domination number of complete graph Kn and cycle Cn. Since ϒ(Kn) = 1 then ϒ (Kn*mK1) = 1 + m. Furthermore, ϒ(Kn*mP2) = 1 + m. Since then . Furthermore, .

Energy and skew-energy of a modified graph

Graph energies draw the greater attention of the scientific community due to their direct applicability in molecular chemistry. In this paper, we establish the energy of a graph obtained by the means of some graph operations. The energy of the product of graph Kn × G, where Kn is a complete graph and G is a simple undirected graph and energy of the corresponding digraph are estimated. Further, the duplication graph DG is considered and proved that the energy E(DG) = 2E(G) and E(DGσ) = 2E(Gσ).

Spectral characterization of the complete graph removing a path: Completing the proof of Cámara–Haemers Conjecture

A graph G is A−DS if every A-cospectral graph of G is isomorphic to G. Denote by Kn∖Pk the graph obtained from the complete graph Kn with n vertices by deleting all edges of a path Pk with k vertices. In 2014, Cámara and Haemers conjectured that Kn∖Pk is A−DS for every 2≤k≤n. The conjecture has been confirmed for k=n (Doob and Haemers, 2002), 2≤k≤6 (Cámara and Haemers, 2014), 7≤k≤9 (Mao et al., 2019) and k≥20 (Liu et al., 2020). In this paper, we completely settle the conjecture by proving the remaining cases.

Recursive constructions of amoebas

Global amoebas are a wide and rich family of graphs that emerged from the study of certain Ramsey-Turán problems in 2-colorings of the edges of the complete graph Kn that deal with the appearance of unavoidable patterns once a certain amount of edges in each color is guaranteed. Indeed, it turns out that, as soon as such coloring constraints are satisfied and if n is sufficiently large, then every global amoeba can be found embedded in Kn such that it has half its edges in each color. Even more surprising, every bipartite global amoeba G is unavoidable in every tonal-variation, meaning that, for any pair of integers r, b such that r + b is the number of edges of G, there is a subgraph of Kn isomorphic to G with r edges in the first color and b edges in the second. The feature that makes global amoebas work are one-by-one edge replacements that leave the structure of the graph invariant. By means of a group theoretical approach, the dynamics of this feature can be modeled. As a counterpart to the global amoebas that “live” inside a possibly large complete graph Kn, we also consider local amoebas which are spanning subgraphs of Kn with the same feature.In an effort to highlight their richness and versatility, we present here three different recursive constructions of amoebas, two of them yielding interesting families per se and one of them offering a wide range of possibilities.

Unions of random trees and applications

In 1986, Janson showed that the number of edges in the union of k random spanning trees in the complete graph Kn is a shifted Poisson distribution. Using results from the theory of electrical networks, we provide a new proof of this result, and we obtain an explicit rate of convergence. This rate of convergence allows us to show a new upper tail bound on the number of trees in G(n,p), for p a constant not depending on n. The number of edges in the union of k random trees is related to moments of the number of spanning trees in G(n,p).As an application, we prove the law of the iterated logarithm for the number of spanning trees in G(n,p). More precisely, consider the infinite random graph G(N,p), with vertex set N and where each edge appears independently with constant probability p. By restricting to {1,2,…,n}, we obtain a series of nested Erdős–Réyni random graphs G(n,p). We show that a scaled version of the number of spanning trees satisfies the law of the iterated logarithm.

On the anti-Ramsey number of forests

We call a subgraph of an edge-colored graph rainbow, if all of its edges have different colors. The anti-Ramsey number of a graph G in a complete graph Kn, denoted by ar(Kn,G), is the maximum number of colors in an edge-coloring of Kn with no rainbow copy of G. In this paper, we determine the exact value of the anti-Ramsey number for star forests and double stars and the approximate value of the anti-Ramsey number for linear forests.

Percolating sets in bootstrap percolation on the Hamming graphs and triangular graphs

The r-neighbor bootstrap percolation on a graph is an activation process of the vertices. The process starts with some initially activated vertices and then, in each round, any inactive vertex with at least r active neighbors becomes activated. A set of initially activated vertices leading to the activation of all vertices is said to be a percolating set. Denote the minimum size of a percolating set in the r-neighbor bootstrap percolation process on a graph G by m(G,r). In this paper, we present upper and lower bounds on m(Knd,r), where Knd is the Cartesian product of d copies of the complete graph Kn which is referred as the Hamming graph. Among other results, when d goes to infinity, we show that m(Knd,r)=1+o(1)(d+1)!rd if r≫d2 and n⩾r+1. Furthermore, we explicitly determine m(L(Kn),r), where L(Kn) is the line graph of Kn also known as triangular graph.

Exact Ramsey numbers of odd cycles via nonlinear optimisation

For a graph G, the k-colour Ramsey number Rk(G) is the least integer N such that every k-colouring of the edges of the complete graph KN contains a monochromatic copy of G. Let Cn denote the cycle on n vertices. We show that for fixed k≥2 and n odd and sufficiently large,Rk(Cn)=2k−1(n−1)+1. This resolves a conjecture of Bondy and Erdős for large n. The proof is analytic in nature, the first step of which is to use the regularity method to relate this problem in Ramsey theory to one in nonlinear optimisation. This allows us to prove a stability-type generalisation of the above and establish a correspondence between extremal k-colourings for this problem and perfect matchings in the k-dimensional hypercube Qk.

Successive shortest paths in complete graphs with random edge weights

Consider a complete graph Kn with edge weights drawn independently from a uniform distribution U(0, 1). The weight of the shortest (minimum‐weight) path P1 between two given vertices is known to be , asymptotically. Define a second‐shortest path P2 to be the shortest path edge‐disjoint from P1, and consider more generally the shortest path Pk edge‐disjoint from all earlier paths. We show that the cost Xk of Pk converges in probability to uniformly for all k ≤ n − 1. We show analogous results when the edge weights are drawn from an exponential distribution. The same results characterize the collectively cheapest k edge‐disjoint paths, that is, a minimum‐cost k‐flow. We also obtain the expectation of Xk conditioned on the existence of Pk.

Monochromatic stars in rainbow [formula omitted]-free and [formula omitted]-free colorings

Given two graphs G and H, we consider the Ramsey-type problem of finding the minimum integer n (denoted by egrk(G:H)) such that n2≥k and for every N≥n, every rainbow G-free k-coloring (using exactly k colors) of the complete graph KN contains a monochromatic copy of H. In this paper, we determine egrk(K3:K1,t) for all integers t≥1 and k≥3 completely. Let S3+ be the unique graph on four vertices consisting of a triangle and a pendant edge. We characterize egrk(S3+:K1,t) for all integers t≥1 and k≥3t−2. We also determine egrk(S3+:K1,t) for integers 1≤t≤5 and k≥4.

Sedentary quantum walks

Let X be a graph with adjacency matrix A. The continuous quantum walk on X is determined by the unitary matrices U(t)=exp⁡(itA) (for t∈R). If X is the complete graph Kn and a∈V(X), then1−|U(t)a,a|≤2/n. Roughly speaking, this means that a quantum walk on a complete graph stays home with high probability. We say that a family of graphs is sedentary if there is a constant c such that 1−|U(t)a,a|≤c/|V(X)| for all t. In this paper we investigate this condition, and produce further examples of sedentary graphs.A cone over a graph X is the graph we get by adjoining a new vertex and making it adjacent to each vertex of X. We prove that if X is the cone over an ℓ-regular graph on n vertices, then |U(t)a,a|≤ℓ2/(ℓ2+4n). It follows that if we choose ℓ and n such that n/ℓ2→0, then a continuous quantum walk starting on the “conical” vertex will remain there with probability close to 1. On the other hand, if ℓ≤2, we show there is a time t such that all entries in the a-column of U(t)ea have absolute value 1/n. We show that there are large classes of strongly regular graphs such that 1−|U(t)a,a|≤c/V(X) for some constant c. On the other hand, for Paley graphs on n vertices, we prove that if t=π/n, then |U(t)a,a|≤1/n.

Monochromatic cycle partitions in random graphs

AbstractErdős, Gyárfás and Pyber showed that every r-edge-coloured complete graph Kn can be covered by 25 r2 log r vertex-disjoint monochromatic cycles (independent of n). Here we extend their result to the setting of binomial random graphs. That is, we show that if $p = p(n) = \Omega(n^{-1/(2r)})$ , then with high probability any r-edge-coloured G(n, p) can be covered by at most 1000r4 log r vertex-disjoint monochromatic cycles. This answers a question of Korándi, Mousset, Nenadov, Škorić and Sudakov.

A combinatorial formula for the Schur coefficients of chromatic symmetric functions

We provide a formula for every Schur coefficient in the chromatic symmetric function of a graph in terms of special rim hook tabloids. This formula is useful in confirming the non-Schur positivity of the chromatic symmetric function of a graph, especially when Stanley’s stable partition method does not work. As applications, we completely characterize Schur positive complete tripartite graphs. We show that any squid graph obtained by attaching n pendent edges to a common vertex on the cycle Cm is not Schur positive if m≠2n−1, and that any pineapple graph obtained by attaching m pendent edges to a common vertex on the complete graph Kn is not Schur positive if n≤2m−2.

Nearly-linear monotone paths in edge-ordered graphs

How long a monotone path can one always find in any edge-ordering of the complete graph Kn? This appealing question was first asked by Chvátal and Komlós in 1971, and has since attracted the attention of many researchers, inspiring a variety of related problems. The prevailing conjecture is that one can always find a monotone path of linear length, but until now the best known lower bound was n2/3-o(1). In this paper we almost close this gap, proving that any edge-ordering of the complete graph contains a monotone path of length n1-o(1).

FCM labeling of some graphs and its line graph

A function f is called an F-centroidal mean labeling of a graph G(V, E) with p vertices and q edges if f : V(G) → {1, 2, 3, …, q + 1} is injective and the induced function g* : E(G) →{1, 2, 3, …, q} defined as for all , is bijective. A graph that admits an F-centroidal mean labeling is called an F-centroidal mean graph. The line graph is one among the graph operations. In this paper, we try to analyse that the line graph operation preserves the F-centroidal mean property for the path Pn , the cycle Cn , the star graph Sn , the complete graph Kn , the graph Pn o S 1, the triangular snake graph Tn and the arbitrary subdivision of S 3.

3-Vertex Friendly Index Set of Graphs

Graph labeling is an assignment of integers to the vertices or the edges, or both, subject to certain conditions. In literature we find several labelings such as graceful, harmonious, binary, friendly, cordial, ternary and many more. A friendly labeling is a binary mapping such that where and represents number of vertices labeled by 1 and 0 respectively. For each edge assign the label , then the function f is cordial labeling of G if and , where and are the number of edges labeled 1 and 0 respectively. A friendly index set of a graph is { runs over all f riendly labeling f of G} and it is denoted by FI(G). A mapping is called ternary vertex labeling and represents the vertex label for . In this article, we extend the concept of ternary vertex labeling to 3-vertex friendly labeling and define 3-vertex friendly index set of graphs. The set runs over all 3 ? vertex f riendly labeling f f or all is referred as 3-vertex friendly index set. In order to achieve , number of vertices are partitioned into such that for all with and la- bel the edge by where . In this paper, we study the 3-vertex friendly index sets of some standard graphs such as complete graph Kn, path Pn, wheel graph Wn, complete bipartite graph Km,n and cycle with parallel chords PCn.