Graph theory

A graph has seven vertices with degrees $3, 3, 3, 3, 2, 2, 2$. How many edges does it have?

A connected plane graph has $10$ vertices and $15$ edges. How many faces does it have (counting the unbounded face)?

What is the largest number of edges in a triangle-free graph on $7$ vertices?

Which of these graphs are planar? (Select all that apply.)

Corollary 5 says: a planar graph with $V$ vertices has $E \le 3V - 6$. Which of the hypotheses are genuinely needed, in the sense that dropping one of them gives a counterexample? (Select all that apply.)

The proof that every red–blue colouring of $K_6$ has a monochromatic triangle starts: "a vertex $v$ has five edges, so three of them share a colour". If we try the same proof on $K_5$, where does it break down?

The Erdős–Szekeres bound says $R(s, t) \le \binom{s + t - 2}{s - 1}$. What upper bound does it give for $R(4, 4)$?

Edges of $K_6$ are unordered pairs s(a, b) : Sym2 (Fin 6), and a red–blue colouring is a function c : Sym2 (Fin 6) → Bool. Which Lean statement says that every colouring has a monochromatic triangle?

A bipartite graph is given as a list adj, where adj[i] is the set of neighbours (in $B$) of vertex $i$ of $A$. Write hall_violator(adj) that returns a smallest set $S \subseteq A$, as a sorted tuple of indices, with $|N(S)| < |S|$, or None if Hall's condition holds (so that, by Hall's theorem, a matching covering $A$ exists).