I believe the book
Hajnal Péter: Gráfelmélet. 1997, Polygon, Szeged.
is an extended answer to exactly this question. (There's a second edition from 2003, but apparently no translations to other languages.)
The writing style of this book makes it accessible to high school students (as opposed to the Lovász book whose concise style makes it ideal for research mathematicians). Thus, most of the material is accessible at high school level, while at the same time the book covers so many difficult topics that it'd be difficult to cover all the proofs even in a semester long high school course.
The book remains a useful reference for BSc combinatorics exams (not alone though, because some necessary topics are missing).
Here are some of the topics included (many of these were mentioned in other responses).
- Flow-cut theorem and algorithm, Menger's theorems, Kőnig-Hall, maximal matching algorithm for bipartite graphs, Tutte's theorem, and even Edmond's algorithm to find maximal matching in any graph.
- Euler circuits, then Dirac's sufficient condition for Hamiltonian circuit
- Graph coloring, Brook's theorem (when a graph's chromatic number reaches its maximum degree), high chromatic graphs without triangle, Hajós's characterisation of graphs with chromatic number (exercise 9.16 in Lovász), Vizing's theorem on edge coloring,
- Turán's theorem, Erdős-Stone theorem about the asymptotic on the number of edges of a graph not containing a particular non-bipartite graph, asymptotic for $ C_4 $-free graphs.
- Ramsey theorems.
- Complexity theory results about graph problems, including Karp reductions between Hamiltonian circuits and chromatic number and independence number. Does not include the Cook-Levin theorem so no problem is actually proven to be NP-complete.
- Planar graphs, duality, Whitney-duals, Euler's theorem, Kuratowski's theorem on the characterization of planar graphs (yes, with a proof), Robertson and Seymour's graph minor theorem without proof, the four color theorem without proof, the five color theorem and Kempe's proof, Hadwiger's conjecture.
- Perfect graphs, Lovász's theorem on the complement of perfect graphs (yes, with a proof), comparability graphs of posets are perfect.
Note finally that before any topic, you'd need to cover some of the first chapter which introduces basic terminology about graphs, which is important to plan if you are giving only a few lectures. (Have you told us about the total length of lectures you are planning to give? give?) Edit: ah, I see the lectures the question is referring to are now in the past, so there's no point to ask such a concrete question.
