136 research outputs found
A hierarchy of randomness for graphs
AbstractIn this paper we formulate four families of problems with which we aim at distinguishing different levels of randomness.The first one is completely non-random, being the ordinary Ramsey–Turán problem and in the subsequent three problems we formulate some randomized variations of it. As we will show, these four levels form a hierarchy. In a continuation of this paper we shall prove some further theorems and discuss some further, related problems
Beyond graph energy: norms of graphs and matrices
In 1978 Gutman introduced the energy of a graph as the sum of the absolute
values of graph eigenvalues, and ever since then graph energy has been
intensively studied.
Since graph energy is the trace norm of the adjacency matrix, matrix norms
provide a natural background for its study. Thus, this paper surveys research
on matrix norms that aims to expand and advance the study of graph energy.
The focus is exclusively on the Ky Fan and the Schatten norms, both
generalizing and enriching the trace norm. As it turns out, the study of
extremal properties of these norms leads to numerous analytic problems with
deep roots in combinatorics.
The survey brings to the fore the exceptional role of Hadamard matrices,
conference matrices, and conference graphs in matrix norms. In addition, a vast
new matrix class is studied, a relaxation of symmetric Hadamard matrices.
The survey presents solutions to just a fraction of a larger body of similar
problems bonding analysis to combinatorics. Thus, open problems and questions
are raised to outline topics for further investigation.Comment: 54 pages. V2 fixes many typos, and gives some new materia
Supersaturation Problem for Color-Critical Graphs
The \emph{Tur\'an function} \ex(n,F) of a graph is the maximum number
of edges in an -free graph with vertices. The classical results of
Tur\'an and Rademacher from 1941 led to the study of supersaturated graphs
where the key question is to determine , the minimum number of copies
of that a graph with vertices and \ex(n,F)+q edges can have.
We determine asymptotically when is \emph{color-critical}
(that is, contains an edge whose deletion reduces its chromatic number) and
.
Determining the exact value of seems rather difficult. For
example, let be the limit superior of for which the extremal
structures are obtained by adding some edges to a maximum -free graph.
The problem of determining for cliques was a well-known question of Erd\H
os that was solved only decades later by Lov\'asz and Simonovits. Here we prove
that for every {color-critical}~. Our approach also allows us to
determine for a number of graphs, including odd cycles, cliques with one
edge removed, and complete bipartite graphs plus an edge.Comment: 27 pages, 2 figure
Recommended from our members
The Chromatic Structure of Dense Graphs
This thesis focusses on extremal graph theory, the study of how local constraints on a graph affect its macroscopic structure. We primarily consider the chromatic structure: whether a graph has or is close to having some (low) chromatic number.
Chapter 2 is the slight exception. We consider an induced version of the classical Turán problem. Introduced by Loh, Tait, Timmons, and Zhou, the induced Turán number ex(n, {H, F-ind}) is the greatest number of edges in an n-vertex graph with no copy of H and no induced copy of F. We asymptotically determine ex(n, {H, F-ind}) for H not bipartite and F neither an independent set nor a complete bipartite graph. We also improve the upper bound for ex(n, {H, K_{2, t}-ind}) as well as the lower bound for the clique number of graphs that have some fixed edge density and no induced K_{2, t}.
The next three chapters form the heart of the thesis. Chapters 3 and 4 consider the Erdős-Simonovits question for locally r-colourable graphs: what are the structure and chromatic number of graphs with large minimum degree and where every neighbourhood is r-colourable? Chapter 3 deals with the locally bipartite case and Chapter 4 with the general case.
While the subject of Chapters 3 and 4 is a natural local to global colouring question, it is also essential for determining the minimum degree stability of H-free graphs, the focus of Chapter 5. Given a graph H of chromatic number r + 1, this asks for the minimum degree that guarantees that an H-free graph is close to r-partite. This is analogous to the classical edge stability of Erdős and Simonovits. We also consider the question for the family of graphs to which H is not homomorphic, showing that it has the same answer.
Chapter 6 considers sparse analogues of the results of Chapters 3 to 5 obtaining the thresholds at which the sparse problem degenerates away from the dense one.
Finally, Chapter 7 considers a chromatic Ramsey problem first posed by Erdős: what is the greatest chromatic number of a triangle-free graph on vertices or with m edges? We improve the best known bounds and obtain tight (up to a constant factor) bounds for the list chromatic number, answering a question of Cames van Batenburg, de Joannis de Verclos, Kang, and Pirot
Extensions of Extremal Graph Theory to Grids
We consider extensions of Turán\u27s original theorem of 1941 to planar grids. For a complete kxm array of vertices, we establish in Proposition 4.3 an exact formula for the maximal number of edges possible without any square regions. We establish with Theorem 4.12 an upper bound and with Theorem 4.15 an asymptotic lower bound for the maximal number of edges on a general grid graph with n vertices and no rectangles
On some interconnections between combinatorial optimization and extremal graph theory
The uniting feature of combinatorial optimization and extremal graph theory is that in both areas one should find extrema of a function defined in most cases on a finite set. While in combinatorial optimization the point is in developing efficient algorithms and heuristics for solving specified types of problems, the extremal graph theory deals with finding bounds for various graph invariants under some constraints and with constructing extremal graphs. We analyze by examples some interconnections and interactions of the two theories and propose some conclusions
A collection of open problems in celebration of Imre Leader's 60th birthday
One of the great pleasures of working with Imre Leader is to experience his
infectious delight on encountering a compelling combinatorial problem. This
collection of open problems in combinatorics has been put together by a subset
of his former PhD students and students-of-students for the occasion of his
60th birthday. All of the contributors have been influenced (directly or
indirectly) by Imre: his personality, enthusiasm and his approach to
mathematics. The problems included cover many of the areas of combinatorial
mathematics that Imre is most associated with: including extremal problems on
graphs, set systems and permutations, and Ramsey theory. This is a personal
selection of problems which we find intriguing and deserving of being better
known. It is not intended to be systematic, or to consist of the most
significant or difficult questions in any area. Rather, our main aim is to
celebrate Imre and his mathematics and to hope that these problems will make
him smile. We also hope this collection will be a useful resource for
researchers in combinatorics and will stimulate some enjoyable collaborations
and beautiful mathematics
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