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Conjectures and Challenges
in
Graph Labelings
Ibrahim Cahit Arkut
Girne American University
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Presentation Plan
• Cordial graphs
• Graceful trees
• Harmonious graphs
• Super magic trees (Beta-
magic)
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Challenges and Conjectures
All trees (arithmetic
progression)
(Lee, 2003)
Friendly index set of
trees
Cordial
All trees
(Cahit, 2000)
Trees with diameter 4Beta-Magic
All trees
(Graham, Sloane,
1980)
Trees with diameter 4Harmonious
All trees
(Ringel-Kotzig,1963)
Trees with diameter 6
and lobsters
Graceful
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Origin of the graph labeling problems
• Problem 25. (Ringel, 1963)
If T is a tree on m edges, then T6K2m+1.
2 3
1
5 2
4 3
1
=K5= 5 x
(Kotzig, 1965)
Cyclically decomposed !
Thm. (Rosa). If T is a graceful tree on m edges,
then T6K2m+1.
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Definition
• Label the vertices/edges of a graph G with
integers from a set S with the property P’ so that
when induce edge/vertex labels computed by
the rule R imposed another property P on the
edge/vertex set.
e.g.,
(a) S={1,2,…,n}, {0,1}, {2,4,…,2k}, {a,b,c,…} or {-1,0,+1} etc.,
(b) P or P’ may be distinct integers, equitably used numbers, etc.,
(c) R addition, modular addition or absolute difference of adjacent
vertex labels, edge sum at a vertex etc.,
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Graceful, Harmonious and Cordial
• f: V T N={0,1,…,e} and f({u,v})=6f(u)-f(v)6,
if induce edge labels N={1,2,…,e} then f is
called a graceful labeling.
• f: V TN=Ze and f({u,v})=f(u)+f(v), if induce
edge labels N=Ze then f is called a
harmonious labeling.
• f: V TN={0,1} and if N={0,1} with
6vf(1)-vf(0) 6b1 and 6ef(1)-ef(0) 6b1 then f is
called a cordial labeling.
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Reason to non-existence results
(according to A. Rosa)
• G has “too many vertices” and “not
enough edges”
• G has “too many edges”
• G has “ the wrong parity”*
Thm. If every vertex has even degree and
6E(G)6ª1,2 (mod 4) then G is not graceful.
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0
1 4
6
4
3
2
0
7
10
15
0 5
4
1
14
12
8
2
3
SOME GRACEFUL
SOME UNGRACEFUL GRAPHS
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Computational results
• All trees at most 27 vertices are graceful.
(Aldred and McKay).
• Decide whether a graph admits a cordial
labeling is NP-complete.
(Cairne and Edwards).
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Probabilistic results
• Almost all random graphs are cordial.
(Godbole, Miller and Ramras).
• Almost all graphs are not graceful.
(Erdös).
• Almost all graphs are not harmonious.
(Graham and Sloane).
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Trees are cordial
• Proof 1. Mathematical
induction on n.
• Proof 2. Horce-race
labeling algorithm
0
1
1 0
1
0
1
0
1
0
1
322110e(1)
221100e(0)
322210v(1)
332111v(0)
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k-equitable labeling of graphs
Cahit (1990)
• For any positive integer k, assign vertex labels
from {0, 1, . . . , k – 1} s.t.
• (1) the number of vertices labeled with i and the
number of vertices labeled with j differ by at
most one and
• (2) the number of edges labeled with i and the
number of edges labeled with j differ by at most
one.
• G(V,E) is graceful if and only if it is |E| + 1-
equitable and is cordial if and only if it is 2-
equitable.
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Some results
• An Eulerian graph with qTk (mod 2k)
edges is not 3-equitable.
• (Szanizló) Cn is k-equitable iff k satisfies:
n∫k; if kª2,3 (mod 4) then n∫k-1
if kª2,3 (mod 4) then nTk (mod 2k).
• (Speyer,Szanizló) All trees are 3 equitable.
Conjecture (Cahit, 1990): All trees are k-
equitable.
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Randomly Cordial Graphs
(Chartrand, Min Lee, Zhang,2002)
• Thm. Connected graph G
of order nr2 is randomly
cordial iff n=3 and G=K3
or n is even and G=K1,n-1
Proof: Very lengthy (17
pages)
n=3
and
K3
K1,n-1
,n=even
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Friendly Index Set of Cordial
Graphs
• f: V(G)TA induces F*:E(G) TA defined
f*(xy)=f(x)+f(y), for each xyeE(G).
• For ieA,
vf(i)=card{veV(G):f(v)=i} and
ef(i)=card{eeE(G):f*(e)=i}.
• Let c(f)={6ef(i)-ef(j)6:(i,j)eAxA}
• f of G is said to be A-friendly if 6vf(i)-vf(j)6b1 for all
(i,j)eAxA
• If c(f) is a (0,1)-matrix for an A-friendly labeling f
then f is said to be A-cordial.
• A=Z2 T FI(G)={6ef(0)-ef(1)6:f is Z2-friendly}
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Friendly index sets of trees
• FI(K1,n)={1} if n is odd and {0,2} otherwise.
• Complete binary tree with depth 1 has
FI={0,2} and FI={0,2,…,2d+1-4} for
depthr2.
• Conjecture (Lee and Ng, 2004). The
numbers in FI(T) for any tree T forms an
arithmetic progression.
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Examples
• Thm. For any G with
q edges
FI(G)m{0,2,…,q} if q is
even
FI(G)m{1,3,…,q} if q is
odd
0 0
1 1
0
1
0
1 1 0
0 1
1
1 1
1
0
011
1
00 1
1 1 0 1 0 1
FI(K1,3
)={1}
FI(K2,2
)={0,4}
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Example (complete binary tree)
{0,1,1,0,0,1}
{0,0,0,1,0,0} {1,1,0,1,1,1}
{0,0,0,0,1,1} {1,1,1,1,1,1} {0,0,0,0,0,0} {1,1,1,1,0,0}
{0,1,1,1,0,1} {1,0,1,1,1,0}
{0,0,0,1,1,1} {1,1,1,0,1,1} {1,1,0,1,1,1} {0,0,1,0,1,1}
e(1)=3,e(0)=3 and e(1)-e(0)=0 FI(T2)={0,2,4}
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Alpha-labeling of graphs (Rosa)
• If f is an graceful
labeling and there
exists mœ{1,2,…,n}
s.t. for an arbitrary
edge (x,y) of G either
f(x)bm and f(y)>m or
f(x)>m and f(y)bm
holds.
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Strong graceful trees
• If f is an graceful
labeling and for every
x,y,zœV(T) with
(x,y),(y,z)œE(T)
either f(x)<f(y) and
f(y)>f(z) or f(x)>f(y)
and f(y)<f(z) holds
then f is called
ordered.
1
7
5
6 2
4
3
Conjecture (Cahit, 1980): All trees are ordered graceful.
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Graceful trees
1
2 3
4
5 6
7
8
9
10
11
12
13
14
All caterpillar trees are graceful
Conjecture (Bermond 1979): All lobster trees are graceful.
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Rotatability
• A graceful tree is called
rotatable if it is possible to
assign smallest label to
an arbitrary vertex.
• Paths and a class of
caterpillars are rotatable. Unrotatable trees
Conjecture:
There exists no trees with
more than two unrotatable
vertices.
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Small rotatable trees
• T(p,1) is rotatable for
pT3,11 (mod 12) and
pT0 (mod 2).
• T(p,r) is rotatable for
p,r=even.
• T(p,r) is not rotatable
for p=odd and r=even.
• T(p,r) is rotatable for
p,r=odd and
(p+r)/2=odd.
p
vertices
r
vertices
T(p,r)
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Spiral canonic labeling
1
10
2
9
3
8
4
7
5
6
1 n 2 n-1 3 n-2
(1) All trees with
diameter four are
graceful (Cahit, 1985)
(2) All trees with
diameter five are
graceful (Hrnciar,
Haviar, 2001)
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Trees with diameter r 6
Rooted trees where the
roots have odd degree
and the lengths of the
paths from the root to the
leaves differ by at most
one and all the internal
vertices have the same
parity.
1
2
3 4
5
6
7 8 9 10
11 131416
17181920
22
2324 25
26
27
28 29 30 31
32 343537
383940
46 45
43
44
3 24 44 23 4 25 42 21
43 22 626 42 215 27 41
7 2841 20
36 15 12 33
n=46,m=2 (mod4)
Classify them as:
(a) TR(o),I(e) ,
(b) TR(o),I(o) ,
(c) TR(e),I(e) ,
(d) TR(e),I(o)
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Harmonious trees
• f is harmonious labeling of G with e edges
if it is possible to label vertices with distinct
elements f(x) of Ze, s.t. f(x,y)=f(x)+f(y).
• If G is a tree exactly one vertex label
repeated.
Conjecture: All trees are harmonious.
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Harmonious labeling of p-stars
• A p-star is a star tree
in which each edge is
a path of length k.
Challenge: Harmonious
labeling of p-stars,
where p=even and
k>2.
10
9
12
8
11
6
75
4
31
20 181614
0
13
191715
mod 20
Harmonious labeling of T(2,10)
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Magic (edge) labeling
• f: V(G)(E(G)T{1,2,…,6V(G)6(6E(G)6} is edge-
magic if f(x)+f(y)+f(xy)=C for any xyeE(G).
• f is called super-magic if additionally
f(V(G))={1,2,…, 6V(G)6} and f(E(G))={6V(G)6+1,…,
6V(G)6(6E(G)6}
• Paths and caterpillars are edge-magic. (Kotzig
and Rosa, 1970).
• Complete binary trees are edge-magic. (Cahit,
1980).
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Alpha-magic (or consecutive magic)
• A(T,v): set of vertices even
distance from v.
• B(T,v): set of vertices odd
distance from v.
• If
f(A(T,v))={1,2,…,6A(T,v)6}
and
f(B(T,v))={6A(T,v)6+1,…,6V(T)6}
then T is called alpha-
magic.
21
3 8
13 18 24 29
1 2 4 5 6 7 9 10
11 12 15 171416 23 2220 19 26 2825 27 30 31
A(T,1)={1,2,3,...,10}
B(T,1)={11,12,....,31}
Complete binary trees are alpha magic
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Beta-magic trees (Cahit 2003)
• A tree T is beta-magic if it is super-magic
and its vertices partitioned into three color
classes C1,C2,C3 of consecutive labels.
• All alpha magic trees are beta magic
(reverse is not true).
• All comets (2-stars) are beta magic.
Conjecture: All trees are beta-magic.
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Mile-stones
• A. Rosa, On certain valuations of the vertices of a
graph, in: Theory of Graphs (Proc. Internat.
Sympos. Rome 1965), Gordon and Breach, N.Y.-
Paris 1967, pp.349-355.
• A. Kotzig and A. Rosa, Magic valuations of finite
graphs, Canadian Math. Bull.,13(4),(1970),451-461.
• R. L. Graham, N.J.A. Sloane, On additive bases and
harmonious graphs, SIAM J. Alg. Disc.
Math.,1(1980), pp.382-404.
• I. Cahit, Cordial graphs: A weaker version of
graceful and harmonious graphs, Ars Combinatoria,
23(1987),pp.201-208.
• J. Gallian, A dynamic survey of graph labeling,
October 2003, (http://www.combinatorics.org), DS6.
