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Another good reference is this Master's thesis of Michael Cavers. Among the results presented are bounds for the clique covering number of graphs which are complements of graphs with very few edges. …

If I understand your questions correctly, I think the answer to (1) is all graphs not containing a cycle as a component, and the answer to (2) is all graphs. Let $G'$ be the graph obtained from a co …

This entry on Cut the Knot gives a proof and the reference to the book Exploring Mathematics with Your Computer by A. Engel. I am not sure if the puzzle was invented by Engel, but hopefully the book …

A colleague of mine recently asked me if this set family had a name (see definition of this below) . I didn't know the answer, so I thought I would consult the MO oracle. Let $\mathcal{S}:=\{ S_1, \ …

Given a graph $G$, the problem of determining the minimum size $\tau(G)$ of a set of edges $X$ such that $G-X$ is triangle-free is indeed NP-complete. This was proved by Yannakakis. Regarding bounds …

Note that the interesting case is if $n \leq U/2$. Otherwise, your bound is worse than the bound $\binom{U}{\lfloor U/2 \rfloor}$ given by Sperner's Theorem, for the size of any antichain on the univ …

I believe your problem is indeed NP-hard, via the following reduction from subset sum. Let $(a_i)_{i=1}^n$ be an instance of SUBSET SUM. We will make an instance of SMALLEST SUBSET SUM as follows. L …

Here's an easy proof in the case that $(h, \ell)=(1, 2k)$, with an explicit bound. Claim. For $(h, \ell)=(1,2k)$, there is an irreducible tiling of a $8k \times 8k$ square. Proof. Tile the $(k,2 …

A counterexample is the planar dual to the Tutte graph. Most of the credit goes to David Eppstein (see the comments below) Let $G$ be the dual to the Tutte graph. Then $G$ is a planar triangulation …

Since it is possible that there are citizens in different states with the exact same valuation of land, we might as well solve the stronger version where each citizen in each state believes that her s …

Yes, in my PhD thesis, we prove the stronger result that for any fixed signed graph $(G, \Sigma)$, there is a polynomial-time algorithm to test if an input signed graph contains a $(G, \Sigma)$-minor. …

There are many non-trivial solutions. By Lagrange's four-square theorem, every natural number can be represented as a sum of $4$ squares (with $0^2$ allowed), and Jacobi's four-square theorem gives t …

Your condition should say that the conflict graph is bipartite instead of non-planar. Let me define the conflict graph more precisely and then give a proof. For a cycle $C$ of $G$ a $C$-path is a pa …

Let $C_1, \dots, C_n$ be a family of disjoint simple curves in a surface $\Sigma$. If $C$ is any simple curve in $\Sigma$, it turns out that we can map $C$ to a curve $C'$ (via a homeomorphism of $\S …

Regarding Question 3, here is a proof that $f(n)=30$ for all $n \geq 40$. Let $H$ be a $2$-connected planar graph with minimum degree $5$. Let $G$ be obtained from $H$ by adding a new vertex inside …