Graph Theory
Related to vertex degrees
R. Inkulu
http://www.iitg.ac.in/rinkulu/
(Related to vertex degrees)
1/7
Vertex degrees and number of edges (review)
• Let G(V, E) be an undirected graph. Then 2|E| =
P
v∈V
deg(v).
• Let G(V, E) be a directed graph. Then
|E| =
P
v∈V
deg− (v) =
(Related to vertex degrees)
P
v∈V
deg+ (v).
2/7
Handshaking lemma
Every undirected simple graph G(V, E) contains an even number of vertices
of odd degree.1
1
In a party, say some people shake hands, an even number of people must have shaken an
odd number of other people’s hands. Hence, the name of the lamma.
(Related to vertex degrees)
3/7
Sperner’s lemma for R2 : an application of handshaking
lemma
Let T be an arbitrary Steiner triangulation of a triangle v1 v2 v3 . Let the vertices of T be colored from the set
{1, 2, 3} such that vi receives the color i (for each i), and only the colors i and j are used for vertices along
the edge from vi to vj (for i 6= j), while the interior vertices are colored arbitrarily with 1, 2 or 3. Then in T
there must be a small tricolored triangle.2
3
1
3
3
3
1
1
3
1
2
2
2
1
1
2
2
except for the vertex corresponding to outer face, every vertex in the subgraph of dual graph (shown above)
has even degree
2
this lemma (extended to Rn ) has important consequence - Brower’s fixed point theorem:
every continuous function f : Bn → Bn of an n-dimensional ball to itself has a fixed point (a
point x ∈ Bn with f (x) = x).
(Related to vertex degrees)
4/7
Tiling rectangles: yet another application of
handshaking lemma
Whenever a rectangle is tiled by rectangles all of which have at least one side
of integer length, then the tiled rectangle has at least one side of integer length.
every black node as well as every circle node (whose x and y coordi are integers) has even degree;
south-west corner of the given rectangle has odd degree
(Related to vertex degrees)
5/7
Degree sequences
The degree sequence of an undirected graph is the sequence of the degrees of
the vertices in non-increasing order.
• For any non-negative non-increasing sequence of integers d1 , d2 , . . . , dn
P
with 1≤i≤n di even, there exists an undirected (not necessarily simple)
graph G with n vertices such that vertex labeled i has degree di for every
1 ≤ i ≤ n.
— homework
• The non-negative non-increasing sequence of integers d1 , d2 , . . . , dn are
the vertex degrees of some loopless undirected graph iff
d1 ≤ d2 + . . . + dn .
P
di is even and
— no proof given
(Related to vertex degrees)
6/7
If a graph G(V, E) has no isolated nodes, then every vertex cover of G is a
dominating set of G. Further, the minimum dominating set size in G is upper
bounded by |V|/2.
— homework
(Related to vertex degrees)
7/7