Chapter 5 Coloring of Graphs

1
Chapter 5Coloring of Graphs
2
5.1 Vertex Coloring and Upper Bound

• Definition
• A k-coloring of a graph G is a labeling fV(G)?S,
  where Sk (or Sk).
• The labels are colors the vertices of one color
  form a color class.
• A k-coloring is proper if adjacent vertices have
  different labels.
• A graph is k-colorable if it has a proper
  k-coloring.
• The chromatic number ?(G) is the least k such
  that G is k-colorable.

3

• Remark
• In a proper coloring, each color class is an
  independent set.
• Graphs with loops are uncolorable multiple edges
  are irrelevant. So we consider simple graph only.

4

• Definition
• A graph G is k-chromatic if ?(G)k.
• A proper k-coloring of a k-chromatic graph is an
  optimal coloring.
• If ?(H)lt ?(G) k for all proper subgraph H of G
  is color-critical or k-critical.
• The clique number of a graph G, written ?(G), is
  the max size of clique in G.

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Proposition 5.1.7

• For every graph G, ?(G)??(G) and ?(G)?n(G)/?(G).
• ?(G) is the max independent set of G.
• ?(G) may exceed ?(G).
• r ? 2, C2r1 join Ks
• ?(G)s2
• ?(G)s3

C5
Ks
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Cartesian Product

• The Cartesian product of G and H, written G?H, is
  the graph with vertex set V(G)?V(H) specified by
  putting (u,v) adjacent to (u,v) iff (1) uu
  and vv?E(H), (2) vv and uu?E(G).

b
H
c
a
x
(x,c)
G
G?H
y
(y,a)
z
(z,c)
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Proposition 5.1.11

• ?(G?H)max?(G), ?(H)
• ?(G?H) ? max?(G), ?(H) k
• f(u,v) g(u)h(v) mod k

2
H
3
1
3
1
1
2
1
G
G?H
2
3
2
3
1
1
2
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Upper Bounds

• Greedy Coloring
• A vertex ordering v1, v2,, vn of V(G), assign vi
  the smallest-indexed color not already used on
  its lower-indexed neighbors.
• ?(G) ? ?(G) 1
• Using greedy coloring to prove.

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• If a graph G has degree sequence d1?d2 ??dn,
  then ?(G) ? 1maxi mindi, i-1
• Apply greedy coloring to the vertices in the
  non-increasing order of degree. Vi has at most
  mindi, i-1 earlier neighbors. We assign the
  color to vi at most 1mindi,i-1. So we maximize
  over i to obtain upper bound.

10
Lemma 5.1.18

• If H is a k-critical graph, then ?(H) ? k-1
• Let x be a vertex of G. H-x is k-1-colorable. If
  d(x)ltk-1, then N(x) cannot use all k-1 color and
  x can be assigned the rest color, contradiction.
• Theorem If G is a graph, then ?(G) ?
  1maxH?G?(H)
• Let k ?(G), and let H be a k-critical subgraph
  of G.
• ?(G)-1 ?(H)-1 ? ?(H) ? maxH?G ?(H)

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Gallai-Roy-Vitaver Theorem

• If D is an orientation of G with longest path
  length l(D), then ?(G)?1l(D). Furthermore,
  equality holds for some orientation of G.

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• Proof
• Let D be an orientation of G. Let D be a maximal
  sub-digraph of D containing no cycle.
• We assign color along the longest path by
  increase 1.
• Every path can be assigned in increasing number.
• So we use 1 l(D) colors.

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• Proof(cont)
• Let e is an edge in D not in D. Since De forms
  a cycle and all path are in increasing order. So
  the two ends of e cannot be the same color.

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• Proof(cont)
• To prove the second statement, we construct D
  st. l(D) ? ?(G)-1.
• Let f be optimal coloring of G. We set uv a
  orient u to v in D iff f(u)ltf(v). So the max
  path length l(D) ? ?(G)-1.

15
Brooks Theorem

• If G is a connected graph other than a complete
  graph or an odd cycle, then ?(G) ? ?(G).
• Proof
• Let G be a connected graph. Let k ?(G) and k ?
  3. Since k1 is a K2, k2 is a odd cycle or
  bipartite.
• Our aim is to order the vertices st. each has at
  most k-1 lower neighbors.

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• Proof(cont)
• If G is not k-regular. We choose a vertex with
  degree not k as vn.
• Choose one neighbor of as vi.
• So every vi has at least higher neighbors. Thus
  vi has at most k-1 lower neighbors.
• So it can be colored by k colors.

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• Proof(cont)
• G is k-regular and G has a cut-vertex x.
• Every component of G-x union x with edges between
  them can be colored by k colors.

x
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• Proof(cont)
• G is k-regular. We assume that G is 2-connected.
• Choose a vn has neighbors v1,v2 such that v1?v2
  and G-v1,v2 is connected.
• We can order G-v1,v2 with 3,,n.
• Every vi before vn has at most k-1 lower neighbor
  and v1,v2 receive the same color. So greedy
  coloring also uses k colors.

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• Proof(cont)
• We now proof that every 2-connected k-regular
  graph with k?3 has such triple v1, v2, vn.
• Choose a vertex x.
• If ?(G-x)2, let v1x, v2 with distance 2 from x
  (G is not complete), and vn is their common
  neighbor.

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• Proof(cont)
• If ?(G-x)1, since G has no cut-vertex, x has a
  neighbor in every leaf block.
• Neighbors v1,v2 of x in two such blocks are
  nonadjacent.
• k ? 3 so G-v1,v2 is connected.

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5.2 Structure of k-chromatic Graphs

• Bound ?(G) ? ?(G) is bad.
• ?(G), ?(G), ?(G) over all graphs is approximating
  to 2ln n, 2ln n, n/(2ln n) (in 8.5), so n/?(G) is
  a good bound.

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Graphs with Large Chromatic Number

• Definition
• A simple graph G, Mycielskis construction
  produces a simple graph Gcontaining G. Beginning
  with G having vertex v1,v2,,vn, add
  Uu1,u2,,un and w. add edges to make ui to
  adjacent to NG(vi), and finally let N(w)U.

v2
u2
v2
w
v1
v1
u1
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• Theorem from a k-chromatic triangle-free graph
  G, Mycielskis construction produces a
  k1-chromatic triangle free graph G.
• ltpfgt V(G)v1,v2,,vn is triangle free, U
  u1,u2,,un is an independent set, and w cannot
  be contained in any triangle.
• So the triangles only can occur on some ui with
  two neighbors of vi, but it contains a triangle
  in G.
• Thus G is triangle-free.

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• ltpf(cont)gt let ?(G) k.
• G can be assigned in k1 colors by set ui the
  same color with vi, and w the color k1. Hence
  ?(G) ? ?(G) 1.
• We now want to prove that ?(G) lt ?(G) .
• If a proper coloring g on G using k colors, let
  g(w)k -gt U using colors 1,2,,k-1, V(G) may
  use k colors.

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• ltpf(cont)gt
• We want to change all color of g(vi) into g(ui)
  st. g is still proper.
• If vi, vj adjacent, since vi also adjacent to uj,
  so vi has different color from vj.
• So G is k-1-colorable.

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Extremal Problems and Turáns Theorem

• Proposition 5.2.5 every k-chromatic graph with n
  vertices at least edges.
• ltpfgt there are pairs of colors meaning the
  two ends of a edge. If (i,j) does not exist,
  color i and j can be merge into one color.
• Maximization is more interesting.

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• Definition
• A complete multipartite graph is a simple graph G
  whose vertices can be partition into Kn1, Kn2,
  Knk. Where u?v iff u, v belong to different Kni.
• We written it as Kn1,n2,,nk.
• Every component in is a complete graph.
• Turán graph Tn,r
• A complete r-partite graph and the vertices
  number m of every part is ?n/r? ? m ? ?n/r?.

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• Lemma Among simple r-partite graphs with n
  vertices, the Turán graph is the unique graph
  with the most edges.
• ltpfgt We consider complete r-partite only.
• let G be a r-partite graph other than Turán graph
  with most edges.
• We chose v from largest class (size i) and move
  it to the smallest class (size j), i-1gtj.
• We loss j edges and gain i-1 edges, so we have
  more edges than G, contradiction.

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• Theorem Among the n-vertex simple graphs with no
  r1-clique, Tn,r has the maximum of edges.
• ltpfgt if we can prove that the maximum is achieved
  by an r-partite graph. Then we can use earlier
  lemma to prove Turán graph is the maximum.
• We want to construct a r-partite graph H from
  graph G with at least as many edges.

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• Prove by induction on r
• When r1 ,then no edges. Consider rgt1.
• Let G as an n-vertex graph with no r1-clique,
  and x?V(G) be a vertex of degree k?(G).
• G is the induced subgraph of G, and V(G)N(x).
  Since G has no r1 clique, G has no r-clique.
• Applying induction hypothesis, there exists a
  r-1-partite H st. e(H)?e(G).

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• Let H be the graph formed by joining N(x) and S
  V(G) - N(x), H is r-partite.
• e(G) ? e(G) (n-k)k ? e(H) (n-k)k e(H)
• Tn,r has the most edges within all r-colorable
  graphs.

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Color-Critical Graphs

• Proposition
• (a) for v?V(G), there is a proper k-coloring of G
  in which the color on v appears on v only, and
  other k-1 appear on N(v).
• (b) for e?E(G), every proper k-1-coloring of G-e
  gives the same color to the two ends of e.
• ltpfgt(a) giving proper k-1-coloring on G-v and
  color k to v forms a proper k-coloring on G. N(v)
  must use k-1 colors, otherwise G is
  k-1-colorable.
• ltpfgt(b) if some k-1-coloring of G-e gave distinct
  colors to the two ends of e, then G is
  k-1-colorable.

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• The join of two color critical graphs is still
  color-critical.

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• Lemma let G be a graph with ?(G)gtk, and let X,Y
  be a partition of V(G). If GX and GY are
  k-colorable, then the edge cut X,Y has at least
  k edges.
• ltpfgt let X1, X2,, Xk and Y1, Y2,, Yk be the
  partition of X and Y by color classes. If there
  are no edges between Xi and Yj, then Xi?Yj is an
  independent set in G.

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• Recall that p.121 3.1.29 for every subgraph of
  Kn,n with more than (k-1)n edges has a matching
  at least k.
• Construct bipartite H with vertices X1,,Xk and
  Y1,,Yk, and edges when no edge on G between Xi
  and Yj.
• Let X,Yltk, then H has more than (k-1)k edges,
  so it has a matching at least k (perfect
  matching).
• We assign the same color to each matching pair.
  Since Xi?Yj matching in H means that they are
  independent set in G, the coloring is proper.

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• Theorem Every k-critical graph is
  k-1-edge-connected.
• ltpfgtusing earlier lemma, proved.

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• Definition
• let S be a set of vertices of G. an S-lobe of G
  is a induced subgraph whose vertex set is S union
  one of the component of G-S.
• Proposition if G is k-critical, then G has no
  cutset x,y with x?y. And there is a S-lobe H
  st. ?(Hxy)k.
• Let Sx,y is a cutset of G with x?y, H1, H2,,
  Ht are S-lobes of G. Each Hi is k-1-colorable. If
  x?y, then we must assign distinct color to x, y
  in each Hi.

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• ltpf(cont)gt thus we can find a coloring st. x, y
  in assigned in the same color in every Hi.
• Then G is k-1-colorable, contradiction.
• Now we prove second statement.
• If for all Hi, ?(Hxy) lt k, then G is
  k-1-colorable.

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Forced Subdivision

• Definition An H-subdivision is a graph obtained
  from a graph H by successive edge subdivisions.

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• Theorem every graph with chromatic number at
  least 4 contains a K4-subdivision.
• ltpfgt induction on n(G)
• When n(G)4, the graph is K4 itself.
• Consider n(G)gt4. G has a 4-critical subgraph H. H
  has no cut-vertex.
• If ?(H)2, let cutset Sx,y and not x?y, then
  there is a S-lobe H st. ?(Hxy)?4.
• Since n(Hxy)ltn(G), we can apply the induction
  hypothesis to obtain K4-subdivision in Hxy.

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• ltpf(cont)gt we replace xy into xy-path in other
  S-lobe other than H, it is a K4-subdivision in
  G. Such path exists since x, y connect to every
  S-lobe.
• If ?(H)3, choose a vertex x. Since H-x is
  2-connected, H-x contains a cycle C. And Since H
  is 3-connected, the Fan lemma (theorem 4.2.23)
  shows that x has 3 vertex-disjoint paths
  connecting to cycle C, it forms a K4-subdivision.

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Enumerative Aspects

• Counting Proper Colorings
• Chordal Graph
• A Hint of Perfect Graphs

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Counting Proper Colorings

• Definition
• Given k?N and a graph G, the value ?(Gk) is the
  number of proper coloring using at most k colors.
• Examples ?(Knk) k(k-1)(k-2)(k-n1), ?( k)
  kn.

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• Proposition 5.3.3 If T is a tree with n
  vertices, then ?(Tk) k(k-1)n-1.
• ltpfgt choosing a vertex as a root and considering
  the coloring from it, then .

k
k-1
k-1
k-1
k-1
k-1
k-1
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• Proposition 5.3.4 let x(r)x(x-1)(x-r1). If
  pr(G) denotes the number of partitions of V(G)
  into r nonempty independent sets, then ?(Gk)
  , which is the polynomial in k of degree
  n(G).
• ltpfgtwhen using r colors in a proper coloring, it
  will partition V(G) into r independent sets,
  which can happen in pr(G) ways.
• When k colors available, k(r) ways to choose
  colors.
• The way to partition V(G) into n(G) independent
  sets is only 1, it leads to the leading term kn.

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• Example
• C4, p10, p21, p32, p41
• ?(C4k)1?k(k-1)2?k(k-1)(k-2)1?k(k-1)(k-2)(k-3)
  k(k-1)(k2-3k3)

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• Theorem if G is a simple graph and e?E(G), then
  ?(Gk) ?(G-ek)- ?(G?ek).
• ltpfgt if the proper coloring ?(G-ek) assigns the
  two ends of e distinct color, then the coloring
  is also proper in ?(Gk).
• If the two ends of e are assigned in the same
  color in ?(G-ek), the number is the same with
  ?(G?ek).
• Example
• ?(C4 k) ?(P3k)- ?(K3k)k(k-1)(k2-3k3)

48

• Theorem the chromatic polynomial ?(Gk) of a
  simple graph G has degree n(G), with integer
  coefficients alternating sign and beginning 1,
  -e(G),.
• ltpfgtwe use induction on e(G). e(G)0 holds.
• ?(G-ek)kn- e(G)-1kn-1 a2kn-2-(-1)iaikn-i
• -?(G?ek) -( kn-1 -
  b1kn-2(-1)i-1bi-1kn-i)
• ?(Gk) kn - e(G)kn-1 (a2b1)kn-2(-1)i
  (aibi-1)kn-i

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• Theorem let c(G) denote the number of components
  of a graph G. Given a set S?E(G) of edges in G,
  let G(S) denote the spanning subgraph of G with
  edge set S. Then the number ?(Gk) of proper
  k-chromatic of G is given by

50

• Example
• Like the theorem of exclusion and inclusion.
• ?(Gk)k4 - 5k3 10k2 - (2k28k1) 5k - k

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• ltpfgt
• Multiple edges do not effect the theorem.
• When all edges have been deleted or contracted,
  the graph remains isolated vertices. The
  remaining vertices corresponding to components of
  G(S)
• So the term is kc(G(S)), and the sign is changed
  by contracting edge, so the contribution is
  positive iff S is even.

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Chordal Graphs

• Definition a vertex of G is simplicial if its
  neighborhood in G is a clique.
• A simplicial elimination ordering is a order
  vn,,v1 for deleting such that when deleting vi,
  vi is a simplicial vertex of the remaining graph
  induced by v1,,vi.

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• Definition
• A chord of a cycle C is an edge not in C whose
  end points lie in C.
• A chordless cycle in G is a cycle of length at
  least 4 that has no chord.
• A graph G is chordal if it is simple and has no
  chordless cycle.

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• Lemma for every vertex x in a chordal graph G,
  there is a simplicial vertex of G among the
  vertices farthest from x in G
• ltpfgt induction on n(G), when n1 trivial.
• If x is adjacent to all other vertices, then G-x
  has a simplicial vertex y. And y in G is also
  simplicial since x adjacent to every vertex.
• We consider the rest case.

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• Let T be the set of farthest points from x, H is
  a component of GT.
• Let S be the set of vertices in G-T having
  neighbors in V(H).
• And let Q be the component of G-S contains x.
• We claim that S is a clique

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• If not, there exist u,v ? S such that u,v being
  not adjacent.
• u,v have neighbors in H, and u,v have neighbors
  in Q. So there is a uv-path through H, and a
  uv-path through Q.
• If u and v non-adjacent, then there is a
  chordless cycle, contradiction.
• So S is a clique.

u
H
Q
v
57

• Let G S?H, we can use induction hypothesis that
  (whether G is a clique or not) there is a u?S
  has a simplicial vertex z ? V(H) farthest from
  it. Since NG(z)?V(G), z is also simplicial in G.
• z is what we want.

58

• Theorem a simple graph has a simplicial
  elimination ordering iff it is a chordal graph.
• -gt let G be a graph with simplicial elimination
  ordering. Let C be a cycle in G of length at
  least 4.
• When we first deleted a vertex v from C.
• Since the neighborhood of v in rest graph is a
  clique, the edge join the two neighbors of v in C
  is a chord of C, so no chordless cycle.

v
59

• lt- by earlier lemma, every chordal graph has a
  simplicial vertex. Since every induced subgraph
  of a chordal graph is a chordal graph, proved.

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A Hint of Perfect graphs

• Definition
• A graph G is perfect if ?(H)?(H) for every
  induced subgraph H?G.
• The clique cover number ?(G) of a graph G is the
  minimum number of cliques in G needed to cover
  V(G) note that ?(G) ?( ).
• A family of graphs G is heredictary if every
  induced subgraph of a graph in G is also a graph
  in G.
• More detail in section 8.1.

61

• Theorem chordal graphs are perfect.
• ltpfgtevery induced subgraph of chordal graph is
  chordal. We only need to prove that ?(G)?(G)
  when G is chordal.
• We have known that G has a simplicial elimination
  ordering, the reverse of the orderingv1,v2,,vn
  .
• For vi, the neighbors of vi among v1,,vi-1
  forms a clique. We apply greedy coloring here.
• If vi uses color k, then 1,,k-1 appear on
  earlier neighbors of vi, and we have a clique
  with size k.
• The obtain a clique whose size equals the number
  of color used.