# Questions tagged [extremal-graph-theory]

Study of graphs satisfying a property that are maximal or minimal with respect to some parameter. A classic example is Turán's Theorem, which exactly characterizes the densest graphs on $n$ vertices without a $K_t$ subgraph.

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### 4-cycles vs eigenvalue information on quasi-random graphs

My (philosophical) question arises from reading the wonderful paper of Chung-Graham-Wilson where the authors introduces the notion of quasi-random graphs. The main purpose of the paper is to show ...
226 views

### Graph metric approximating Euclidean metric

I've been reading Wolfram's recent articles about graph/mesh/grid structures as an analogy for physical space, and it seems to me that there will be a problem getting the notion of distance to work ...
53 views

### Expected number of bridges in a random subgraph

I am researching connectivity in random subgraphs and have come across the following problem. A bridge between two vertices $a$ and $b$ of a graph $G$ is an edge $e$ such that removing $e$ from $G$ ...
260 views

### Sum of degree differences for simple graphs

For a simple graph $G$ on $n$ vertices, let us define $$\mathcal{I}_{n}(G)=\sum_{i,j=1}^{n}|\deg\ x_{i}-\deg\ x_{j}|^{3}.$$ I know that there are many different topological indices defined and ...
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### Density of bipartite $d$-degenerate graph

A graph $G$ is $d$-degenerate if every subgraph of $G$ contains a vertex of degree at most $d$. It is known that an $n$-vertex $d$-degenerate graph has at most $d(n-1)$ edges. However, if we are given ...
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### At what aspect ratio does the Ruzsa-Szemeredi Theorem begin?

One of the many equivalent phrasings of the Ruzsa-Szemeredi theorem is as follows. Suppose one has a three-layered $n$-node graph $G = (V=L_1 \cup L_2 \cup L_3, E)$, and one can partition $E$ into ...
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### Matching book embedding of Cartesian products of graphs

In the book embedding of a graph $G$ , each vertex of $G$ is placed on the spine and each edge is placed in the pages without crossing each other edge. If vertices have degree at most one in each ...
206 views

### Population of P people, where each person knows K others, how many people mutually know each other

If you have a population of $P$ people, where each person knows $K$ others within the population (does not have to be mutual, i.e., if I know you, you don't necessarily know me), and $1<K<P$, ...
53 views

### A simple equality for book embedding of two graphs

A book embedding of a graph $G$ consists of placing the vertices of $G$ on a spine and assigning edges of the graph to pages so that edges in the same page do not cross each other. The page number is ...
164 views

### Lovasz local lemma for the edge model

In order to successfully apply the Lovasz local lemma, one needs the events to be relatively independent. This (sometimes) works well in the $G(n,p)$ model of random graphs, where the presence or ...
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### The maximum number of edges in an even-cycle-free graph with $n$ vertices

Problem Given any positive integer $n$, what is the maximum number of edges in an even-cycle-free graph with $n$ vertices? Is the above problem an unsolved problem in extremal graph theory? Are there ...
185 views

### Graph properties that imply a bounded number of edges

Many combinatorial problems can be reduced to bounding the number of edges in a given graph with $n$ vertices. Each time I encounter such a problem, I check whether the corresponding graph has a ...
A graph $G=(V,E)$ is said to be vertex-critical if removing a vertex $v\in V$ reduces the chromatic number $\chi(\cdot)$. Edge-criticality is defined in a similar manner. Moreover, $G$ is called ...