Deepak Bal
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View article: On the Number of Monochromatic Cliques in a Graph
On the Number of Monochromatic Cliques in a Graph Open
Nordhaus and Gaddum proved sharp upper and lower bounds on the sum and product of the chromatic number of a graph and its complement. Over the years, similar inequalities have been shown for a plenitude of different graph invariants. In th…
View article: Edge-coloring $K_{n, n}$ with no 2-colored $C_{2k}$
Edge-coloring $K_{n, n}$ with no 2-colored $C_{2k}$ Open
The generalized Ramsey number $r(G, H, q)$ is the minimum number of colors needed to color the edges of $G$ such that every isomorphic copy of $H$ has at least $q$ colors. In this note, we improve the upper and lower bounds on $r(K_{n, n},…
View article: Exploring the Dynamics of Lotka-Volterra Systems: Efficiency, Extinction Order, and Predictive Machine Learning
Exploring the Dynamics of Lotka-Volterra Systems: Efficiency, Extinction Order, and Predictive Machine Learning Open
For years, a main focus of ecological research has been to better understand the complex dynamical interactions between species which comprise food webs. Using the connectance properties of a widely explored synthetic food web called the c…
View article: Nordhaus-Gaddum inequalities for the number of cliques in a graph
Nordhaus-Gaddum inequalities for the number of cliques in a graph Open
Nordhaus and Gaddum proved sharp upper and lower bounds on the sum and product of the chromatic number of a graph and its complement. Over the years, similar inequalities have been shown for a plenitude of different graph invariants. In th…
View article: Generalized Ramsey numbers of cycles, paths, and hypergraphs
Generalized Ramsey numbers of cycles, paths, and hypergraphs Open
Given a $k$-uniform hypergraph $G$ and a set of $k$-uniform hypergraphs $\mathcal{H}$, the generalized Ramsey number $f(G,\mathcal{H},q)$ is the minimum number of colors needed to edge-color $G$ so that every copy of every hypergraph $H\in…
View article: A lower bound on the multicolor size-Ramsey numbers of paths in hypergraphs
A lower bound on the multicolor size-Ramsey numbers of paths in hypergraphs Open
The r-color size-Ramsey number of a k-uniform hypergraph H, denoted by Rˆr(H), is the minimum number of edges in a k-uniform hypergraph G such that for every r-coloring of the edges of G there exists a monochromatic copy of H. In the case …
View article: On the multicolor Ramsey numbers of balanced double stars
On the multicolor Ramsey numbers of balanced double stars Open
The balanced double star on $2n+2$ vertices, denoted $S_{n,n}$, is the tree obtained by joining the centers of two disjoint stars each having $n$ leaves. Let $R_r(G)$ be the smallest integer $N$ such that in every $r$-coloring of the edges…
View article: Large monochromatic components in expansive hypergraphs
Large monochromatic components in expansive hypergraphs Open
A result of Gyárfás [12] exactly determines the size of a largest monochromatic component in an arbitrary $r$ -colouring of the complete $k$ -uniform hypergraph $K_n^k$ when $k\geq 2$ and $k\in \{r-1,r\}$ . We prove a result which says tha…
View article: Large monochromatic components in hypergraphs with large minimum codegree
Large monochromatic components in hypergraphs with large minimum codegree Open
A result of Gyárfás says that for every 3‐coloring of the edges of the complete graph , there is a monochromatic component of order at least , and this is best possible when 4 divides . Furthermore, for all and every ‐coloring of the edges…
View article: A lower bound on the multicolor size-Ramsey numbers of paths in hypergraphs
A lower bound on the multicolor size-Ramsey numbers of paths in hypergraphs Open
The $r$-color size-Ramsey number of a $k$-uniform hypergraph $H$, denoted by $\hat{R}_r(H)$, is the minimum number of edges in a $k$-uniform hypergraph $G$ such that for every $r$-coloring of the edges of $G$ there exists a monochromatic c…
View article: Larger matchings and independent sets in regular uniform hypergraphs of high girth
Larger matchings and independent sets in regular uniform hypergraphs of high girth Open
In this note we analyze two algorithms, one for producing a matching and one for an independent set, on $k$-uniform $d$-regular hypergraphs of large girth. As a result we obtain new lower bounds on the size of a maximum matching or indepen…
View article: Large monochromatic components in expansive hypergraphs
Large monochromatic components in expansive hypergraphs Open
A result of Gyárfás exactly determines the size of a largest monochromatic component in an arbitrary $r$-coloring of the complete $k$-uniform hypergraph $K_n^k$ when $k\geq 2$ and $r-1\leq k\leq r$. We prove a result which says that if one…
View article: Large monochromatic components in hypergraphs with large minimum codegree
Large monochromatic components in hypergraphs with large minimum codegree Open
A result of Gyárfás says that for every $3$-coloring of the edges of the complete graph $K_n$, there is a monochromatic component of order at least $\frac{n}{2}$, and this is best possible when $4$ divides $n$. Furthermore, for all $k\geq …
View article: The Matching Process and Independent Process in Random Regular Graphs and Hypergraphs
The Matching Process and Independent Process in Random Regular Graphs and Hypergraphs Open
In this note, we analyze two random greedy processes on sparse random graphs and hypergraphs with a given degree sequence. First we analyze the matching process, which builds a set of disjoint edges one edge at a time; then we analyze the …
View article: Full Degree Spanning Trees in Random Regular Graphs
Full Degree Spanning Trees in Random Regular Graphs Open
We study the problem of maximizing the number of full degree vertices in a spanning tree $T$ of a graph $G$; that is, the number of vertices whose degree in $T$ equals its degree in $G$. In cubic graphs, this problem is equivalent to maxim…
View article: Rainbow spanning trees in randomly coloured $G_{k-out}$
Rainbow spanning trees in randomly coloured $G_{k-out}$ Open
Given a graph $G=(V,E)$ on $n$ vertices and an assignment of colours to its edges, a set of edges $S \subseteq E$ is said to be rainbow if edges from $S$ have pairwise different colours assigned to them. In this paper, we investigate rainb…
View article: New Lower Bounds on the Size-Ramsey Number of a Path
New Lower Bounds on the Size-Ramsey Number of a Path Open
We prove that for all graphs with at most $(3.75-o(1))n$ edges there exists a 2-coloring of the edges such that every monochromatic path has order less than $n$. This was previously known to be true for graphs with at most $2.5n-7.5$ edges…
View article: Zero-forcing in random regular graphs
Zero-forcing in random regular graphs Open
The zero forcing process is an iterative graph colouring process in which at each time step a coloured vertex with a single uncoloured neighbour can force this neighbour to become coloured.A zero forcing set of a graph is an initial set of…
View article: The Bipartite $K_{2,2}$-Free Process and Bipartite Ramsey Number $b(2, t)$
The Bipartite $K_{2,2}$-Free Process and Bipartite Ramsey Number $b(2, t)$ Open
The bipartite Ramsey number $b(s,t)$ is the smallest integer $n$ such that every blue-red edge coloring of $K_{n,n}$ contains either a blue $K_{s,s}$ or a red $K_{t,t}$. In the bipartite $K_{2,2}$-free process, we begin with an empty graph…
View article: A note on a candy sharing game
A note on a candy sharing game Open
Suppose k students sit in a circle and are each distributed some initial amount of candy. Each student begins with an even amount of candy, but their individual amounts may vary. Upon the teacher’s signal, each student passes half of their…
View article: Hamiltonian Berge cycles in random hypergraphs
Hamiltonian Berge cycles in random hypergraphs Open
In this note we study the emergence of Hamiltonian Berge cycles in random r -uniform hypergraphs. For $r\geq 3$ we prove an optimal stopping time result that if edges are sequentially added to an initially empty r -graph, then as soon as t…
View article: A Ramsey property of random regular and k‐out graphs
A Ramsey property of random regular and k‐out graphs Open
In this study we consider a Ramsey property of random ‐regular graphs, . Let be fixed. Then w.h.p. the edges of can be colored such that every monochromatic component has order . On the other hand, there exists a constant such that w.h.p.,…
View article: Zero Forcing Number of Random Regular Graphs
Zero Forcing Number of Random Regular Graphs Open
The zero forcing process is an iterative graph colouring process in which at each time step a coloured vertex with a single uncoloured neighbour can force this neighbour to become coloured. A zero forcing set of a graph is an initial set o…
View article: Packing Tree Factors in Random and Pseudo-Random Graphs
Packing Tree Factors in Random and Pseudo-Random Graphs Open
For a fixed graph H with t vertices, an H-factor of a graph G with n vertices, where t divides n, is a collection of vertex disjoint (not necessarily induced) copies of H in G covering all vertices of G. We prove that for a fixed tree T on…
View article: A greedy algorithm for finding a large 2‐matching on a random cubic graph
A greedy algorithm for finding a large 2‐matching on a random cubic graph Open
A 2‐matching of a graph G is a spanning subgraph with maximum degree two. The size of a 2‐matching U is the number of edges in U and this is at least where n is the number of vertices of G and κ denotes the number of components. In this ar…
View article: Issue information
Issue information Open
Aims It is the aim of this journal to meet two main objectives: to cover the latest research on discrete random structures, and to present applications of such research to problems combinatorics and computer science.The goal is to provide …
View article: Rainbow perfect matchings and Hamilton cycles in the random geometric graph
Rainbow perfect matchings and Hamilton cycles in the random geometric graph Open
Given a graph on n vertices and an assignment of colours to the edges, a rainbow Hamilton cycle is a cycle of length n visiting each vertex once and with pairwise different colours on the edges. Similarly (for even n ) a rainbow perfect ma…