Hamilton 1
Denitions
cycle - a simple cycle that includes every vertex with
out repetition.
graph
isomorphism - a dierent representation of a graph that is the
exact same.
simple graph - no loops, no more than one edge between 2 vertices.
isomorphic graphs
bipartite graph - 2-colorable graph, can be split into two disjoint sets so every vertex in set 1 connects to a vertex in set
2.
vertex coloring - coloring of vertices of a graph such that no 2
adjacent vertices are the same color (4 is the largest minimum)
plane graph
chromatic number - number of colors you need.
planar graph - 2-D graph no overlaps.
chromatic polynomial - the number of ways a graph can be colored using no more than a given number of colors. (t is the
number of colors).
subdivision of a graph - when you subdivide edges in the graph.
complete graph - every edge that can be drawn is.
Kn = (t)(t 1)(t (n 1))
complete bipartite graph - 2 disjoint sets of vertices where every
vertex in set 1 is connected to every vertex in the second set.
Tree with n vertices: (t)(t 1)(n1)
Cycle Cn = (t 1)n + (1)n (t 1)
vertex - The dot.
edge - the line
tree - an undirected graph with any 2 vertices connected by
EXACTLY 1 simple path. A connected graph with out cycles.
multiple edge - more than 1 edge between the 2 same vertices.
forest - disjoint union of trees.
loop - edge that starts and ends at the same vertex.
shortest path - shortest path...
degree - The number of edges drawn to a vertex.
degree sequence - the non-increasing sequence of vertex degrees
in a graph
path - in a sequence of vertices such that from each vertex there
is an edge to the next vertex in the sequence.
trail - A walk in which all the edges are distinct.
walk - alternating sequence of vertices and edges, starting and
ending with a vertex.
cycle - a path that starts and ends at the same vertex.
circuit - a closed trail.
closed walk - same really as a cycle.
Eulerian circuit - an Euler trail that starts and ends on the
same vertex.
Eulerian trail - a trail that visits each edge once.
minimum spanning tree -tree of smallest weight.
2
Theorem
Handshaking lemma - deg (v ) = 2 E .
Prims algorithm. Pick a vertex. Pick the least weight vertex. Keep picking the least weight vertex until done. (vertex
focused)
Kruskals algorithm. Pick the least weighted edges. keep doing
so until you have all the vertices connected.
Eulers Formula. V E + F = 2.
Kuratowskis theorem: A graph is planar IFF it doesnt contain
a subgraph of K5 or K3,3 .
Dijkstras shortest path algorithm: Uh, just look at all the
paths and nd the smallest one.