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MATH 350
GRAPH THEORY & COMBINATORICS
PROF. SERGEY NORIN, FALL 2013
Contents
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Basic definitions
Connectivity
Trees
Spanning Trees
Shortest paths
Eulerian & Hamiltonian cycles
Bipartite Graphs
Matchings in Bipartite Graphs
Menger’s Theorem and Separations
Digraphs and Network flows
Ramsey’s Theorem
Matchings and Tutte’s Theorem
Vertex coloring
Edge coloring
Series-Parallel Graphs
Planar graphs
Kuratowski’s Theorem
Coloring planar graphs
Perfect graphs
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1. Basic definitions
Definition. A graph G consists of a set of vertices V (G), a set of edges E(G) and an
incidence relation on E(G), namely every edge is incident to 1 or 2 vertices, called its ends.
Definition. A loop is an edge with one end.
Definition. Two edges are parallel if they have the same ends.
Definition. A simple graph has o loops or parallel edges.
Definition. A null graph is the unique graph with no vertices or edges.
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Definition. A complete graph on n vertices is a simple graph on n vertices in which every
pair of vertices is adjacent. It is denoted Kn .
Definition. A path with n edges Pn is a graph whose vertices and edges can be enumerated
{v1 , v2 , . . . , vn+1 and {e1 , e2 , . . . , en }, respectively, so that ei has ends vi and vi+1 , for all
1 ≤ i ≤ n.
Definition. A cycle of length n Cn is a graph whose vertices and edges can be enumerated
{v1 , v2 , . . . vn } and {e1 , e2 , . . . , en } such that ei has ends vi and vi+1 for 1 ≤ i ≤ n − 1 and en
has ends vn and v1 .
Definition. The degree of a vertex v ∈ V (G) is the number of edges incident to it (with
loops counted twice).
Definition. The incidence matrix of a graph G is a matrix with rows indexed by the
vertices of G and columns by edges of G.
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Lemma 1.1 (Handshaking Lemma). In any graph G, v∈V (G) deg(v) = 2|E(G)|.
Corollary 1.2. In any graph G, the number of vertices of odd edgree is even.
Definition. We say that a graph H is a subgraph of a graph G if V (H) ⊆ V (G), E(H) ⊆
E(G) and every edge of H has the same ends in H as it does in G. We write H ⊆ G is H
is a subgraph of G.
Definition. We say that two graphs G and H are isomorphic if there exists a bijection
between V (G) and V (H) and a bijection between E(G) and E(H) which preserve incidence.
If G ∼
= H, then G and H in fact look the same up to relabelling.
2. Connectivity
Definition. A walk in a graph G is a sequence v0 , e1 , v1 , e2 , . . . , vn−1 , en , vn such that vi ∈
V (G), ei ∈ E(G), 1 ≤ i ≤ n. The edge ei has ends vi−1 and vi for 1 ≤ i ≤ n. These edges
are not necesarrily distinct, so we allow them to repeat.
Definition. A graph G is said to be connected if for any pair u, v ∈ V (G), there exists a
walk in G from u to v.
Definition. A partition of a set Z is a pair {X, Y }, X, Y ⊆ Z, X ∪ Y = Z, X ∩ Y = ∅.
Lemma 2.1. If G is not connected, then there exists a partition {X, Y } of V (G) such that
X, Y 6= ∅ and no edge of G has one end in X and another in Y .
Lemma 2.2. Let G be a connected graph. Then, for every X ⊆ V (G), X 6= ∅, X 6= V (G),
some edge of G has one end in X and another V (G) − X.
Lemma 2.3. If there exists a walk from u to v in G, then there is a path.
Lemma 2.4. Let H1 , H2 be connected subgraphs of a graph G. Then, if H1 ∩ H2 is not a
null graph, then H1 ∪ H2 is connected.
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Definition. A connected component of a graph G is a maximal non-null connected
subgraph of G.
Corollary 2.5. Every vertex v in G is contained in a unique connected component. If u, v
are in the same connected component and v, w are in the same connected component, then
so are u, w.
Definition. We say that H is obtained from G by deleting one edge e ∈ E(G) if V (H) =
V (G), E(H) = E(G) − {e} and H is a subgraph of G. We denote it by H = G\e. Such
definition can be applied to vertices in a similar fashion.
Definition. An edge e of G is a cut-edge if it does not belong to any cycle in G.
Theorem 2.6. Let G be a graph, e ∈ E(G) an edge with ends u, v and H = G\e. Denote
by comp(G) the number of connected component of G. Theb, (i) either e is not a cut-edge,
u, v belongs to the same connected component of H and comp(H) = comp(G); (ii) e is a
cut-edge of G, u, v belong to distinct component of H and comp(H) = comp(G) + 1.
3. Trees
Definition. A forest is a graph with no cycles (forests are simple).
Definition. A tree is a non-null connected forest.
Lemma 3.1. Let G be a forest. Then, comp(G) = |V (G)| − |E(G)|.
Corollary 3.2. If G is a tree, then |E(G)| = |V (G)| − 1.
Definition. A leaf in a graph G is a vertex of degree 1.
Theorem 3.3. Let T be a tree with |V (T )| ≥ 2. Let X be the set of all leaves of T and Y
be the set of all verties of T with degree greater than or equal to 3. Then, |X| ≥ |Y | + 2.
Lemma 3.4. Let T be a tree, |V (T )| ≥ 2. Then, T has at least 2 leaves and if T has exactly
2 leaves u and v, then T is a path from u to v.
Lemma 3.5. Let T be a tree and let v ∈ V (T ) be a leaf. Then, T \v is a tree.
Lemma 3.6. Let T be a tree and u, v ∈ V (T ). Then, there exists a unique path in T from
u to v.
4. Spanning Trees
Definition. We say that T is a spanning tree in a graph G if T ⊆ G, T is a tree and
V (T ) = V (G).
Lemma 4.1. Let G be a non-null connected graph. Let H be a subgraph of G chosen so
that V (H) = V (G), H is connected and H is minimal with these properties. Then G is a
spanning tree of G.
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Remark. The complete graph Kn has nn−2 spanning trees.
Lemma 4.2. Let G be a connected, non-null graph. Let H ⊆ G be chosen so that H is
acyclic and maximal subject to this. Then H is a spanning tree of G.
Definition. Let G be a graph, T be a spanning tree of G and let f ∈ E(G) − E(T ). Let
C be a cycle in G such that f ∈ E(C) and E(C) − {f } ⊆ E(T ). Then, C is called the
fundamental cycle of f with respect to T .
Lemma 4.3. Let T be a spanning tree of a graph G and let f ∈ E(G) − E(T ). Then, G
contains a unique fundamental cycle of f with respect to T .
Lemma 4.4. Let T be a spanning tree of G, f ∈ E(G) − E(T ) and C be the fundamental
cycle of f with respect to T . Further let e ∈ E(C) − {f }. Then, the graph T 0 obtained from
the graph T \e by adding f is a spanning tree of G.
Definition. Let G be a graph and let w :P
E(G) → R (we may assume that w : E(G) → R+ ).
Let T be a spanning tree of G such that e∈E(T ) w(e) is minimum among all spanning trees
of F . Then, we call T a min-cost tree for G and w.
Lemma 4.5. Let G be a graph, w : E(G) → R, T a min-cost tree and let f, C, e be as in
Lemma 4.4. Then, w(e) ≤ w(f ).
Theorem 4.6. Let G be a graph, w : E(G) → R, T a min-cost tree with respect to w
and G. Let e1 , e2 , . . . , en−1 be all the edges of T , where n = |V (G)|, and suppose that
w(e1 ) ≤ w(e2 ) ≤ · · · ≤ w(en−1 ). Then, for every 1 ≤ i ≤ n − 1, ei is an edge of G with
minimal weight subject to the fact that {e1 , e2 , . . . , ei−1 , ei } does not contain cycle (an edge
set of a cycle in G) and ei ∈
/ {e1 , e2 , . . . , ei−1 }.
5. Shortest paths
Definition. Let G be a connected graph. Let s, t ∈ V (G) and w : E(G)P→ R+ be the
“length function”. We say P is a shortest path from s to t if w(P ) = e∈E(P ) w(e) is
minimum among all paths from s to t in G.
Definition. Let distG (u, v) denote the length of the shortest path from u to v. We say that
a tree T ⊆ G is a shortest path tree for s ∈ V (G) if s ∈ V (T ) and distG (s, u) = distT (s, u),
for every u ∈ V (T ). In particular, T need not be spanning.
Theorem 5.1. Let G, w and s be as above. Let T be the shortest path tree for s. Assume
that V (T ) 6= V (G). Then, among all the edges of G with one end in V (T ) and another
in V (G) − V (T ), select an edge f with ends x ∈ V (T ) and y ∈ V (G) − V (T ) such that
distT (s, x) + w(f ) is minimum. Let T 0 be the tree obtained from T by adding edge f and
vertex y. Then, T 0 is a shortest path tree for s.
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6. Eulerian & Hamiltonian cycles
Definition. An Eulerian cycle in a graph G is a closed walk which uses every edge in G
exacly once.
Theorem 6.1 (Euler). A graph G with minimum degree 1 has an Eulerian cycle if and only
if the degree of all vertices is even and it is connected.
Lemma 6.2. Let G be a graph with at least one edge and no vertex of degree 1. Then G
contains a cycle.
Lemma 6.3. Let G be a graph such that the degree of every vertex is even. Then, there
exists a collection of cycles C1 , C2 , . . . , CK in G such that every edge of F belong to exactly
one cycle in the collection.
Definition. We say that a cycle C ⊆ G is Hamiltonian if V (C) = V (G).
Note. There is no nice, necessary and sufficient condition for a graph to have a Hamiltonian
cycle.
Lemma 6.4. Let G be a graph, X ⊆ V (G) with X 6= ∅ such that comp(G\X) > |X|. Then,
G has no Hamiltonian cycle.
Theorem 6.5 (Dirac-Pos´a). Let G be a simple graph on n vertices such that deg(u) +
deg(v) ≥ n for all pair of non-adjacent vertices u, v ∈ V (G). Then, G has a Hamiltonian
cycle.
Corollary 6.6. Let G be a simple graph on n vertices with n ≥ 3 and suppose that either
(i) |E(G)| ≥ n2 − n + 3 or (ii) every vertex of G has degree at least d n2 e. Then G has an
Hamiltonian cycle.
7. Bipartite Graphs
Definition. We say that (A, B) is a bipartition of a graph G if A ∪ B = V (G), A ∩ B = ∅
and every edge of G has one end in A and another in B. A graph is bipartite if it has a
bipartition.
Lemma 7.1. (i) A graph G is bipartite if and only if every component of it is bipartite. (ii)
Every tree is bipartite.
Theorem 7.2. Let G be a graph. Then, the following are equivalent:
(i) G is bipartite.
(ii) G contains no closed walk of odd length.
(iii) G contains no odd cycle.
Definition. We say that a subgraph H of G is induced if every edge of G with both ends
in V (H) belongs to H.
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Theorem 7.3. Let G be a simple graph. Then, the following are equivalent:
(i) G is bipartite.
(ii) G contains no induced odd cycle.
8. Matchings in Bipartite Graphs
Definition. A matching in G is a subset M ⊆ E(G) so that no edge is a loop and no two
edges in M are incident to the same edge.
Definition. A path P in G is M -alternating if the edges of P alternately belong to M
and E(G) − M . P is M -augmenting if it also satisfies that the endpoints are not incident
to edges in M .
Theorem 8.1 (Berge). Let M be a matching of G. There is a matching M 0 with |M 0 | ≥ |M |
if and only if there is an M -augmenting path.
Definition. The matching number of G, denoted ν(G) is the maximum size of a matching
in G.
Theorem 8.2 (K¨onig). Let G be a bipartite graph and k ∈ N. Then, the following are
equivalent:
(i) G has a matching M with |M | ≥ k.
(ii) There is no subset X ⊆ V (G) with |X| < k such that X meets every edge of G.
Definition. Let X ⊆ V (G) and e ∈ E(G). We say that e is covered by X if e is incident
to v ∈ X. If X covers all e ∈ E(G), we say that X is a vertex cover.
Definition. A matching M is perfect if it covers all vertices in the graph. In particular,
|M | = |V (G)|/2.
Definition. A subset X ⊆ V (G) is a vertex cover of G if every edge of G has an end in
X.
Definition. The vertex cover number of G, denoted τ (G) is the minimum size of a vertex
cover in G.
Corollary (K¨onig). If G is bipartite, then ν(G) = τ (G).
Theorem 8.3. If G is a bipartite and every vertex has the same degree d > 0, then G has
a perfect matching.
Theorem 8.4 (Hall). Let G be bipartite with bipartition (A, B). Then, the following are
equivalent:
(i) There exists a matching in G covering A.
(ii) For all Γ ⊆ A, vertices in Γ have at least |Γ| neighbours in B.
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9. Menger’s Theorem and Separations
Theorem 9.1. For Q, R ⊆ V (G), k ∈ N, exactly one of the following holds.
(i) There exists paths P1 , P2 , . . . , Pk each with one end in Q and another in R, pairwise
vertex disjoint.
(ii) There exists a spearation (A, B), |A ∩ B| < k with Q ⊆ A and R ⊆ B.
Theorem 9.2 (Menger). Let s, t ∈ V (G) be distinct, non-adjacent vertices in a graph G
and k ≥ 0. Then, exactly one of the following holds.
(i) There exists paths P1 , P2 , . . . , Pk from s to t, all disjoint except for their ends.
(ii) There exists a separation (A, B) of G of order < k such that s ∈ A − B, t ∈ B − A.
Definition. A graph G is k -connected if |V (G)| ≥ k + 1 and G\X is connected for all
X ⊆ V (G) with |X| < k.
Theorem 9.3. Let G be a k-connected graph, s, t ∈ V (G) distinct. Then, there exists paths
P1 , P2 , . . . , Pk from s to t in G, all disjoint except for their ends.
Definition. Let G be a graph, X ⊆ V (G). Then δ(X) denotes the set of all edges of G with
one end in X and another in V (G) − X.
Theorem 9.4. Let k ≥ 0 be an integer, G a graph, s, t ∈ V (G) distinct vertices of G. Then,
exactly one of the following holds.
(i) There exists paths P1 , P2 , . . . , Pk in G from s to t pairwise edge-disjoint, that is, E(Pi )∩
E(Pj ) = ∅ for i 6= j.
(ii) There exists X ⊆ V (G), s ∈ X, t ∈ V (G) − X and |δ(X)| < k.
Definition. Let G be a graph. A line graph of G, denoted L(G) is a graph with
V (L(G)) = E(G) and e, f ∈ E(G) are adjacent in L(G) if and only if they share an end.
10. Digraphs and Network flows
Definition. A digraph G is a graph with every edge e given a direction, that is, on end of
e is chosen as its head and the other as its tail.
Definition. An edge is said to be directed from its tail to its head.
Definition. A directed path in a digraph from s to t is a path from s to t in which the
tail of every edge preceeds its head as we traverse the path from s to t.
Definition. For X ⊆ V (G), let δ + (X) denote the set of all edgs with the tail in X and the
head in V (G) − X. Let δ − (X) := δ + (V (G) − X).
Lemma 10.1. Let G be a digraph, s, t ∈ V (G) distinct. Then, exactly one of the following
holds.
(i) There exists a directed path from s to t.
(ii) There exists X ⊆ V (G) such that s ∈ X, t ∈ V (G) − X and δ + (X) = ∅.
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Definition. Let G be a digraph, s, t ∈ V (G) be distinct vertices of G. The function φ :
E(G) → R+ is called an s-t flow if
X
X
φ(e) =
φ(e)
e∈δ + ({v})
e∈δ − ({v})
for all v ∈ V (G) − {s, t}.
Definition. The value of φ is
X
φ(e) −
e∈δ + (s)
X
φ(e).
e∈δ − (s)
Lemma 10.2. Let G be a digraph, φ an s-t flow of value k. Then, for all X ⊆ V (G) with
s ∈ X and t ∈ V (G) − X,
X
X
φ(e) −
φ(e) = k.
e∈δ + (X)
e∈δ − (X)
Lemma 10.3. Let φ be an integral s-t flow of value k. Then there exist directed paths
P1 , P2 , . . . , Pk from s to t in G and every edge e of G belongs to at most φ(e) paths.
Definition. Let G be a digraph, s, t be distinct vertices of G. For every e ∈ E(G), let
c(e) ∈ Z+ be the capacity of this edge. An s-t flow φ is c-admissible if φ(e) ≤ c(e) for all
e ∈ E(G).
Definition. A path P with end in s and another in some vertex v is called an augmenting
path for φ if
(i) φ(e) ≤ c(e) − 1 for every edge e which is used in the forward direction as P is traversed
from s to v.
(ii) φ(e) ≥ 1 if e ∈ E(P ) is traversed in the opposite direction.
Lemma 10.4. Let G, s, t, c be as above. Let φ be an integral c-admissible s-t flow. If there
exists an augmenting path P for φ from s to t, then there is a c-admissible s-t flow of value
larger than the value of φ.
Theorem 10.5 (Max-Flow Min-Cut — Berge-Fulkerson). Let G be a directed graph, s, t ∈
V (G) distinct, c a capacity function and k ≥ 0. Then, exactly one of the following holds.
(i) There exists a c-admissible s-t flow of total value k.
P
(ii) There exists X ⊆ V (G), s ∈ X, t ∈ V (G) − X such that e∈δ+ (X) c(e) < k.
11. Ramsey’s Theorem
Definition. A subset X ⊆ V (G) is called stable if no edge has both ends in X, that is, no
loops at vertices of X are allowed.
Definition. A subset X ⊆ V (G) is called a clique if every two edges in X is adjacent. In
other words, a clique is a complete subgraph of G.
Definition. Let s, t ≥ 1 be integers. The Ramsey number R(s, t) is the smallest integer
n such that every simple graph on n vertices either has an independent set of size s of a
clique of size t.
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Remark. R(1, k) = R(k, 1) = 1 and R(2, k) = R(k, 2) = k.
Theorem 11.1 (Ramsey’s Theorem, Erd˝os-Szekeres). The number R(s, t) exists for all
s, t ≥ 1 and R(s, t) ≤ R(s, t − 1) + R(s − 1, t) for s, t ≥ 2.
Corollary 11.2. For s, t ≥ 1, R(s, t) ≤ s+t−2
.
s−1
Definition. A coloring of a set S in k colors is a map c : S → {1, 2, . . . k}. c(s) for s ∈ S
is called the color of s.
Definition. A multicolor Ramsey number Rk (s1 , s2 , . . . , sk ) for k, s1 , . . . , sk 1 ≥ 1 all
integers is the minimum n such that for every coloring of edges of Kn in k colors, there exists
1 ≤ i ≤ k and a complete graph Kn on si vertices so that all edges of the subgraph have
color i.
Theorem 11.3. For all k, s1 , . . . , sk ≥ 1, Rk (s1 , s2 , . . . , sk ) exists.
Theorem 11.4 (Schur). For all k ∈ N, there exists n ∈ N such that in every coloring of
{1, 2, . . . , n} in k colors, one can find integers x, y, z of the same color so that x + y = z.
Theorem 11.5 (Schur). For every n ≥ 1, there exists p0 ≥ 1 such that for all prime p ≥ p0 ,
there exists integers x, y, z not divisible by p such that xn + y n ≡ z n (mod p).
Theorem 11.6 (Erd˝os). For s even, R(s, s) ≥ 2s/2 .
12. Matchings and Tutte’s Theorem
Theorem 12.1 (Tutte). A graph G has a perfect matching if and only if compo (G\X) ≤ |X|
for every X ⊆ V (G).
Theorem 12.2 (Tutte-Berge formula). Let G be a graph. Then, G has a matching of size
k ∈ Z+ if and only if compo (G\X) ≤ |X| + |V (G)| − 2k.
Theorem 12.3. Let G be a 3-regular graph (that is, deg(v) = 3 for all v ∈ V (G)). If G has
no cut-edge, then G has a perfect matching.
13. Vertex coloring
Definition. For a graph G, a proper k -vertex coloring of G is a map φ : V (G) → S for
a set S with |S| = k such that if u, v are the ends of an edge e ∈ E(G), φ(u) 6= φ(v). In
particular, no k-coloring exists for a graph containing loops for any k.
Definition. The chromatic number χ(G) is the minimum k such that there exists a
k-vertex coloring of G.
Definition. The clique number, denoted ω(G), is the maximum number if vertices in a
complete subgraph of G
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Definition. The independence number, denoted α(G), is the maximum size of an independant set in G.
Lemma 13.1. Let G be a loopless graph. Then,
(i) χ(G) ≥ ω(G).
(ii) χ(G) ≥ |V (G)|/α(G).
Algorithm (Greedy coloring algorithm). Input: A graph G and an ordering (v1 , v2 , . . . , vn )
of vertices. Output: The resulting proper coloring of G.
• Start by coloring vi in color 1.
• Once {v1 , v2 , . . . , vn } are colored, color vi+1 in the smallest available color (that is,
the smallest positive integer wich is not used to color any of the neighbors of vi
among {v1 , v2 , . . . , vi }.
• Continue until all the vertices are colored.
Definition. The maximum degree of a vertex in G is denoted by ∆(G).
Definition. A graph G is k -degenerate if every subgraph of G has a vertex degree less
than k. For instance, trees are 1-degenerate.
Lemma 13.2. Let G be a k-degenerate loopless graph. Then, χ(G) ≤ k + 1. In particular,
χ(G) ≤ ∆(G) + 1.
Theorem 13.3 (Brooks). Let G be a connected, loopless graph that is not complete and
not an odd cycle. Then, χ(G) ≤ ∆(G).
14. Edge coloring
Definition. For a loopless graph G, a k -edge coloring is a map φ : E(G) → S with |S| = k
such that φ(e) 6= φ(f ) for e, f incident to the same vertex.
Definition. The edge chromatic number, denoted χ0 (G) is the smallest k such that G
admits a k-edge coloring. In particular, χ0 (G) = χ(L(G)).
Note. ∆(G) ≤ χ0 (G) ≤ 2∆(G) − 1.
Lemma 14.1. Let G be a graph with ∆(G) ≤ d for some d. Then, G is a subgraph of some
d-regular graph H. Moreover, if G is loopless (or bipartite), then H can be chosen to be
loopless (resp. bipartite).
Theorem 14.2 (K¨onig). If G is bipartite, then χ0 (G) = ∆(G).
Definition. A 2-factor in a graph in a graph G is a collection F ⊆ E(G) such that every
vertex is incident with exactly 2 edges.
Lemma 14.3. Let G be a loopless 2k-regular graph. Then, E(G) can be partitioned into k
2-factors.
Theorem 14.4 (Shannon). Let G be a loopless graph. Then χ0 (G) ≤ d3∆(G)/2e
Theorem 14.5 (Vizing). If G is simple, then χ0 (G) ≤ ∆(G) + 1.
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15. Series-Parallel Graphs
Definition (Edge contraction). Let e be an edge of a graph G with ends u, v. The graph
G/e obtained by contracting e in G is produced by deleting e and identifying u and v.
Definition. A graph H is a minor of a graph G if it can be obtained from a subgraph of
G by contracting edges sequentially.
Conjecture (Hadwiger’s Conjecture). Let G be a loopless graph with no Kt minor. Then,
χ(G) ≤ t − 1. (Open problem for t ≥ 7).
Lemma 15.1. Every 3-connected graph G contains K4 as a minor.
Lemma 15.2. Let G be a simple graph with no K4 minor. Let X ⊆ V (G) be a clique with
|X| ≤ 2 and X 6= V (G). Then, there exists v ∈ V (G) − X with deg(v) ≤ 2.
Corollary 15.3. If G is a simple graph with no K4 minor, then it is 2-degenerate. Therefore,
G is 3-colorable. Hence, Hadwiger’s Conjecture holds for t = 4.
Definition. A graph is series-parallel if it can be constructed from an empty graph by
repeatedly
• Adding a vertex of degree less than 1.
• Adding a loop or a parallel edge.
• Subdividing an edge.
Lemma 15.4. A minor of a series-parallel graph is series-parallel.
Lemma 15.5. A graph G is series parallel if and only if it has no K4 minor.
16. Planar graphs
Definition. A drawing of a graph in the plane satisfies the following.
• Vertices are represented by distinct points on the plane.
• Edges are represented by simple curves joining their corresponding ends.
• These curves do not intersect each other or themselves.
• These curves do no pass through except for their ends.
Definition. A graph is planar if it admits a planar drawing.
Definition. A drawing divides the plane into regions where two points of the plane not
belonging to the drawing belong to the same region if and only if there is a simple curve
from one point to another disjpint from the drawing. We denote by Reg(G) the number of
regions of a planar graph G.
Theorem (Jordan Curve Theorem). Every simple closed curve divides the plane into 2
regions.
Lemma 16.1. If G is a graph drawn in the plane and e ∈ E(G) is not a cut-edge, then the
regions on the different sides of e are different.
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PROF. SERGEY NORIN, FALL 2013
Lemma 16.2. If F is a forest drawn in the plan, then Reg(F ) = 1.
Theorem 16.3 (Euler’s formula). Let G be a graph drawn in the plane. Then,
|V (G)| − |E(G)| + Reg(G) = 1 + comp(G).
Definition. If R is a region of a drawing of G, then the length of a region R, denoted
Length(R) is the number of edges on the boundary or R, with edge e counted twice if R lies
on both sides of e.
Lemma 16.4. If G is a simple connected graph drawn in the plane with |E(G)| ≥ 2, then
Length(R) ≥ 3 for all region R of the drawing of G.
Theorem 16.5. Let G be a planar simple graph such that |E(G)| ≥ 2.
|E(G)| ≤ 3|V (G)| − 6 and if G has no K3 subgraphs, then |E(G)| ≤ 2|V (G)| − 4.
Then,
Definition. A complete bipartite graph Km,n is a simple graph with bipartition (A, B) such
that |A| = m and |B| = n, |E(Km,n )| = mn. That is, every vertex of A is adjacent to every
vertex of B.
Corollary 16.6. The graph K5 and K3,3 are non-planar.
17. Kuratowski’s Theorem
Lemma 17.1. Let G be a 2-connected loopless graph drawn in the plane. Then, the boundary of every region is a cycle in G.
Lemma 17.2. Let C be a cycle, X, Y ⊆ V (C). Then, exactly one of the following holds.
(i) There exists u, v ∈ V (C) such that if P, Q are two paths from u to v in C comprising
the cycle, then X ⊆ V (P ) and Y ⊆ V (Q).
(ii) There exists distinct x1 , y1 , x2 , y2 ∈ V (C) appearing in this cyclic order such that
xi ∈ X and yi ∈ Y for i = 1, 2.
(iii) X = Y and |X| = |Y | = 3.
Theorem 17.3 (Kuratowski’s Theorem). A graph G is planar if and only if it contains
neither K5 nor K3,3 as a minor.
Definition. G is called a subdivision of H if G is obtained from H by subdividing edges,
that is, every edge of H is replaced by a path with the same length. If G is a subdivision of
H, then H is a minor of G (the converse does not always hold).
Theorem 17.4. A graph G is planar if and only if G does not contain a subdivision of K5
or K3,3 as a subgraph.
MATH 350
GRAPH THEORY & COMBINATORICS
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18. Coloring planar graphs
Lemma 18.1. Let G be a simple planar graph with at least 3 vertices. Then,
X
(6 − deg(v)) ≥ 12.
v∈V (G)
Corollary 18.2. If G is a planar and simple graph, then G is 5-degenerate. In particular,
χ(G) ≤ 6.
Theorem 18.3 (Heawood). If G is a planar and loopless graph, then χ(G) ≤ 5.
Theorem 18.4 (Appel-Haken — The Four Color Theorem). If G is planar and loopless,
then χ(G) ≤ 4.
Definition. A planar triangulation is a planar graph in which every region, including the
infinite one, is a triangle.
Lemma 18.5. Let G be a simple planar triangulation. Then, G contains one of the following.
(i) A vertex of degree less than 4.
(ii) Two neighboring vertices of degree 5.
(iii) A region with vertices of degree 5, 6 and 6.
Theorem 18.6. Let G be a planar triangulation which contains the subgraph drawn in the
figure (see class notes) such that all the triangular regions in the figure are the drawings of
G. Let G0 be obtained from G by deleting the vertices {x, y, z, w}, identifying a and c and
adding an edge between e and f . If G0 is 4-colorable, then so is G.
Theorem. For every simple 3-regular planar graph G, χ0 (G) = 3.
Definition. Let G be a connected graph drawn on the plane and let G∗ be another graph
drawn in the plane. We say that G∗ is the planar dual graph of G if
(i) Every region of G contains exactly one vertex of G∗ .
(ii) Every edge of G is crossed exactly one edge of G∗ and the drawings of G and G∗ are
otherwise disjoint.
(iii) |E(G)| = |E(G∗ )|.
Theorem 18.7 (Tait). Let G be a planar triangulation and let G∗ be a planar dual of G.
Then, χ(G) ≤ 4 if and only if χ0 (G∗ ) = 3.
19. Perfect graphs
Definition. We say that a graph G is perfect if χ(H) = ω(H) for every induced subgraph
H of G. In other words, G is perfect if the chromatic number of H is equal to the clique
number of H for any subgraph H.
Definition. Let G be a simple graph. A complement G of G is a simple graph with
V (G) = V (G) and two vertices in G are adjacent if and only if they are not adjacent in G.
Lemma 19.1. Complements of bipartite graphs are perfect.
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PROF. SERGEY NORIN, FALL 2013
Lemma 19.2. Line graphs of bipartite graphs are perfect.
Lemma 19.3. Complements of line graph of bipartite graphs are perfect.
Definition. A chordal graph is a graph H that has no cycles of length at least 4 as induced
subgraphs.
Definition. We say that a graph G is obtained by gluing subgraphs G1 and G2 along a
subgraph S if G = G1 ∪ G2 and G1 ∩ G2 = S.
Theorem 19.4. A graph is chordal if and only if it can be obtained by repeatedly gluing
along complete subgraphs starting with complete graphs.
Corollary 19.5. Chordal graphs are perfect.
Definition. Let G be a simple graph and x ∈ V (G). Let G0 be obtained from G by adding
a new vertex x0 and joining it by edges to x and all the niehgbors of x. We say that G0 is
obtained from G by expanding x to an edge xx0 .
Lemma 19.6. If G is perfect and G0 is obtained from G by expanding x ∈ V (G) to an edge
xx0 , then G0 is perfect.
Theorem 19.7 (Lov´asz — Weak Perfect Graph Theorem). G is perfect if and only if G is
perfect.
Theorem 19.8 (Chudnovsky-Robertson-Seymour-Thomas). G is perfect if and only if ut
dies bit cibtaub C2k+1 or C2k+1 for k ≥ 2 as an induced subgraph.