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Consider the (simple, undirected) complete graph $K_n$. Is it possible to partition its edges into less than $n$ cliques, each having less than $n$ vertices? Note: It is easy to see that $n$ cliques …

After some search, I have found out that a solution was already published in 1948. De Bruijn and Erdos proved that it is not possible to partition the edges of $K_n$ into fewer than $n$ smaller clique …

Let $Q$ be a hereditary class of graphs. (Hereditary = closed with respect to taking induced subgraphs.) Let $Q_n$ denote the set of $n$-vertex graphs in $Q$. Let us say that $Q$ contains almost all g …

Let us say that a graph class has the jump property, if it either contains all $n$-vertex graphs for every large enough $n$, or else the fraction of $n$-vertex graphs that belong to the class approach …

The network topology of wireless ad hoc networks and wireless sensor networks are often captured by a random geometric graph. This means, picking random points in a planar domain, and connecting any t …

Consider an undirected graph $G=(V,E)$ with non-negative edge costs. Given an integer $k$ with $0\leq k\leq |E|$, let us call an edge set $C\subseteq E$ a $k$-discounted cut, if the following hold: …

(This is to answer Sasho Nikolov's comment, but it is way too long for the comment field, so I post it as an answer.) The two examples in the original question are special cases of LP duality. There …

Instance: An undirected graph $G$ with two distinguished vertices $s\neq t$, and an integer $k\geq 2$. Question: Does there exist an $s-t$ path in $G$, such that the path touches at most $k$ vertices …

It is well known that many important graph parameters show (strong) concentration on random graphs, at least in some range of the edge probability. Some typical examples are the chromatic number, maxi …

What is the complexity of the following problem? Instance: Simple, undirected graph $G$, and a positive integer $k$. Question: Does $G$ have an induced subgraph on at least $k$ vertices, such that a …

There are quite a few theorems, mostly in graph theory and combinatorial optimization, that are often referred to as good characterizations. They typically put a property in $NP\cap co-NP$, by showing …

Given a simple undirected graph $G$, find a subset $A\neq \emptyset$ of vertices, such that for any vertex $x\in A$ at least half of the neighbors of $x$ are also in $A$, and the size of $A$ is mi …

Let $G$ be an undirected graph, given with a planar drawing. We do not assume $G$ is a planar graph, we just fix a planar representation of it, such that each vertex is represented by a distinct poi …

There are some results/concepts in TCS which are used in areas other than the "home area" where they emerged. For example, NP-completeness has complexity theory as its home area, but it is also used i …

What you look for in question Q1 is known as an $f$-factor of the graph. Here $f$ is a non-negative integer valued function on the vertices, $f(v)$ specifying the degree we want in the subgraph at ver …