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It might be possible to prove that such graphs are perfect without first observing that they are Berge and applying the SPG Theorem, but I suspect this would be hard. It has been noted that the prope …

For every tree on $n$ vertices, there are exactly $4 \times 3^{n-1}$ $4$-colorings. This follows from the fact that its chromatic polynomial is $t(t-1)^{n-1}$. (Or by induction since every tree wit …

As has already been pointed out, this is the chromatic number $\chi$. (For example, the assertion that all planar graphs have $ \chi \le 4$ is the famous "Four color theorem.") You say your graph has …

Lovasz's topological lower bounds on chromatic number were extended by Babson and Kozlov -- they have a series of articles, all available on the arXiv. A nice place to start is "Complexes of Graph Hom …

Define a graph $G_1$ where the vertices of $G_1$ are the points of the plane $\mathbb{R}^2$, and a pair of vertices $p, q$ is connected by an edge if and only if the Euclidean distance $d(p,q) =1$. Th …

A notorious problem in combinatorics is the following: If we color $\mathbb{R}^2$ so that no pair of points at unit distance get the same color, what is the fewest number of colors required? This numb …