Catalogue Search | MBRL
Are you sure you want to remove the book from the shelf?
{{itemTitle}}
25,742
result(s) for
"mathematics - combinatorics"
Sort by:
Secret history : the story of cryptology
\"Codes are a part of everyday life, from the ubiquitous Universal Price Code (UPC) to postal zip codes. They need not be intended for secrecy. They generally use groups of letters (sometimes pronounceable code words) or numbers to represent other words or phrases. There is typically no mathematical rule to pair an item with its representation in code. A few more examples will serve to illustrate the range of codes\"-- Provided by publisher.
Book
On the genera of polyhedral embeddings of cubic graph
In this article we present theoretical and computational results on the existence of polyhedral embeddings of graphs. The emphasis is on cubic graphs. We also describe an efficient algorithm to compute all polyhedral embeddings of a given cubic graph and constructions for cubic graphs with some special properties of their polyhedral embeddings. Some key results are that even cubic graphs with a polyhedral embedding on the torus can also have polyhedral embeddings in arbitrarily high genus, in fact in a genus close to the theoretical maximum for that number of vertices, and that there is no bound on the number of genera in which a cubic graph can have a polyhedral embedding. While these results suggest a large variety of polyhedral embeddings, computations for up to 28 vertices suggest that by far most of the cubic graphs do not have a polyhedral embedding in any genus and that the ratio of these graphs is increasing with the number of vertices.
Journal Article
Exponential multivalued forbidden configurations
The forbidden number$\\mathrm{forb}(m,F)$ , which denotes the maximum number of unique columns in an$m$ -rowed$(0,1)$ -matrix with no submatrix that is a row and column permutation of$F$ , has been widely studied in extremal set theory. Recently, this function was extended to$r$ -matrices, whose entries lie in$\\{0,1,\\dots,r-1\\}$ . The combinatorics of the generalized forbidden number is less well-studied. In this paper, we provide exact bounds for many$(0,1)$ -matrices$F$ , including all$2$ -rowed matrices when$r > 3$ . We also prove a stability result for the$2\\times 2$identity matrix. Along the way, we expose some interesting qualitative differences between the cases$r=2$ ,$r = 3$ , and$r > 3$ .
Journal Article
Domination in Kn\\\odel Graphs
Given a graph and an integer $k$, it is an NP-complete problem to decide whether there is a dominating set of size at most $k$. In this paper we study this problem for the Kn\"odel Graph on $n$ vertices using elementary number theory techniques. In particular, we show an explicit upper bound for the domination number of the Kn\"odel Graph on $n$ vertices any time that we can find a prime number $p$ dividing $n$ for which $2$ is a primitive root.
Journal Article
Wiener Index and Remoteness in Triangulations and Quadrangulations
Let$G$be a a connected graph. The Wiener index of a connected graph is the sum of the distances between all unordered pairs of vertices. We provide asymptotic formulae for the maximum Wiener index of simple triangulations and quadrangulations with given connectivity, as the order increases, and make conjectures for the extremal triangulations and quadrangulations based on computational evidence. If$\\overline{\\sigma}(v)$denotes the arithmetic mean of the distances from$v$to all other vertices of$G$ , then the remoteness of$G$is defined as the largest value of$\\overline{\\sigma}(v)$over all vertices$v$of$G$ . We give sharp upper bounds on the remoteness of simple triangulations and quadrangulations of given order and connectivity.
Journal Article
Determining Genus From Sandpile Torsor Algorithms
We provide a pair of ribbon graphs that have the same rotor routing and Bernardi sandpile torsors, but different topological genus. This resolves a question posed by M. Chan [Cha]. We also show that if we are given a graph, but not its ribbon structure, along with the rotor routing sandpile torsors, we are able to determine the ribbon graph's genus.
Journal Article
The LexCycle on$\\overline{P_{2}\\cup P_{3}}$ -free Cocomparability Graphs
A graph$G$is a cocomparability graph if there exists an acyclic transitive orientation of the edges of its complement graph$\\overline{G}$ . LBFS $^{+}$is a variant of the generic Lexicographic Breadth First Search (LBFS), which uses a specific tie-breaking mechanism. Starting with some ordering$\\sigma_{0}$of$G$ , let$\\{\\sigma_{i}\\}_{i\\geq 1}$be the sequence of orderings such that$\\sigma_{i}=$ LBFS $^{+}(G, \\sigma_{i-1})$ . The LexCycle( $G$ ) is defined as the maximum length of a cycle of vertex orderings of$G$obtained via such a sequence of LBFS $^{+}$sweeps. Dusart and Habib conjectured in 2017 that LexCycle( $G$ )=2 if$G$is a cocomparability graph and proved it holds for interval graphs. In this paper, we show that LexCycle( $G$ )=2 if$G$is a$\\overline{P_{2}\\cup P_{3}}$ -free cocomparability graph, where a$\\overline{P_{2}\\cup P_{3}}$is the graph whose complement is the disjoint union of$P_{2}$and$P_{3}$ . As corollaries, it's applicable for diamond-free cocomparability graphs, cocomparability graphs with girth at least 4, as well as interval graphs. Comment: 11 pages, 9 figures
Journal Article
Algorithmics of matching under preferences
Matching problems with preferences are all around us: they arise when agents seek to be allocated to one another on the basis of ranked preferences over potential outcomes. Efficient algorithms are needed for producing matchings that optimise the satisfaction of the agents according to their preference lists. In recent years there has been a sharp increase in the study of algorithmic aspects of matching problems with preferences, partly reflecting the growing number of applications of these problems worldwide. The importance of the research area was recognised in 2012 through the award of the Nobel Prize in Economic Sciences to Alvin Roth and Lloyd Shapley.
eBook
Extremal problems of double stars
In a generalized Turán problem, two graphs$H$and$F$are given and the question is the maximum number of copies of$H$in an$F$ -free graph of order$n$ . In this paper, we study the number of double stars$S_{k,l}$in triangle-free graphs. We also study an opposite version of this question: what is the maximum number edges/triangles in graphs with double star type restrictions, which leads us to study two questions related to the extremal number of triangles or edges in graphs with degree-sum constraints over adjacent or non-adjacent vertices.
Journal Article
Antifactors of regular bipartite graphs
Let$G=(X,Y;E)$be a bipartite graph, where$X$and$Y$are color classes and$E$is the set of edges of$G$ . Lovász and Plummer LoPl86 asked whether one can decide in polynomial time that a given bipartite graph$G=(X,Y; E)$admits a 1-anti-factor, that is subset$F$of$E$such that$d_F(v)=1$for all$v\\in X$and$d_F(v)\\neq 1$for all$v\\in Y$ . Cornuéjols CHP answered this question in the affirmative. Yu and Liu YL09 asked whether, for a given integer$k\\geq 3$ , every$k$ -regular bipartite graph contains a 1-anti-factor. This paper answers this question in the affirmative.
Journal Article