Topic
Diamond graph
About: Diamond graph is a research topic. Over the lifetime, 31 publications have been published within this topic receiving 344 citations.
Papers
Posted Content•
TL;DR: In this paper, it was shown that there exist arbitrarily large n-point sets such that any D-embedding of X into the diamond graph of Newman and Rabinovich requires distortion at least k(p-1 + 1) + 1.
Abstract: We show that any embedding of the level-k diamond graph of Newman and Rabinovich into $L_p$, $1 < p \le 2$, requires distortion at least $\sqrt{k(p-1) + 1}$. An immediate consequence is that there exist arbitrarily large n-point sets $X \subseteq L_1$ such that any D-embedding of X into $\ell_1^d$ requires $d \geq n^{\Omega(1/D^2)}$. This gives a simple proof of the recent result of Brinkman and Charikar which settles the long standing question of whether there is an $L_1$ analogue of the Johnson-Lindenstrauss dimension reduction lemma.
103 citations
TL;DR: The problem of counting tilings of a plane region using specified tiles can be recast as the problem of finding (perfect) matchings of some subgraph of an Aztec diamond graph An, or more generally calculating the sum of the weights of all the matchings, where the weight of a matching is equal to the product of the (pre-assigned) weights of the constituent edges as discussed by the authors.
96 citations
TL;DR: In this paper, the authors considered the diamond graph and showed that the final size of the graph produced by adding edges uniformly at random, provided no such edge creates a copy of the complete graph K4.
Abstract: Let K4- denote the diamond graph, formed by removing an edge from the complete graph K4. We consider the following random graph process: starting with n isolated vertices, add edges uniformly at random provided no such edge creates a copy of K4-. We show that, with probability tending to 1 as ni¾?∞, the final size of the graph produced is i¾?logi¾?ni¾?n3/2. Our analysis also suggests that the graph produced after i edges are added resembles the uniform random graph, with the additional condition that the edges which do not lie on triangles form a random-looking subgraph. © 2013 Wiley Periodicals, Inc. Random Struct. Alg., 45, 513-551, 2014
21 citations
1 Sep 2019
TL;DR: The graceful chromatic number is found, the minimum k for which a graph has a graceful k-coloring, for some well-known graphs and classes of graphs, such as diamond graph, Petersen graph, Moser spindle graph, Goldner-Harary graph, friendship graphs, fan graphs and others.
Abstract: Graph colorings are a major area of study in graph theory involving the constrained assignment of labels (colors) to vertices or edges. There are many types of colorings defined in the literature. The most common type of coloring is the proper vertex k-coloring which is defined as a vertex coloring from a set of k colors such that no two adjacent vertices share a common color. Our central focus in this paper is a variant of the proper vertex k-coloring problem, termed graceful coloring introduced by Gary Chartrand in 2015. A graceful k-coloring of an undirected connected graph G is a proper vertex coloring using k colors that induces a proper edge coloring, where the color for an edge (u,v) is the absolute value of the difference between the colors assigned to vertices u and v. In this work we find the graceful chromatic number, the minimum k for which a graph has a graceful k-coloring, for some well-known graphs and classes of graphs, such as diamond graph, Petersen graph, Moser spindle graph, Goldner-Harary graph, friendship graphs, fan graphs and others.
10 citations
Posted Content•
TL;DR: This analysis suggests that the graph produced after i edges are added resembles the uniform random graph, with the additional condition that the edges which do not lie on triangles form a random‐looking subgraph.
Abstract: Let K_4^- denote the diamond graph, formed by removing an edge from the complete graph K_4. We consider the following random graph process: starting with n isolated vertices, add edges uniformly at random provided no such edge creates a copy of K_4^-. We show that, with probability tending to 1 as $n \to \infty$, the final size of the graph produced is $\Theta(\sqrt{\log(n)} \cdot n^{3/2})$. Our analysis also suggests that the graph produced after i edges are added resembles the random graph, with the additional condition that the edges which do not lie on triangles form a random-looking subgraph.
10 citations
Related Topics (5)
Performance Metrics
| Year | Papers |
|---|---|
| 2021 | 3 |
| 2020 | 4 |
| 2019 | 5 |
| 2018 | 1 |
| 2017 | 2 |
| 2016 | 3 |