On k-Edge-magic Cubic Graphs

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On k-Edge-magic Cubic Graphs
Sin-Min Lee, San Jose State University Hsin-hao
Su, Stonehill College Yung-Chin Wang, Tzu-Hui
Institute of Technology 24th MCCCC At Illinois
State University September 11, 2010
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Supermagic Graphs

• For a (p,q)-graph, in 1966, Stewart defined that
  a graph labeling is supermagic iff the edges are
  labeled 1, 2, 3, , q so that the vertex sums are
  a constant.

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k-Edge-Magic Graphs

• A (p,q)-graph G is called k-edge-magic (in short
  k-EM) if there is an edge labeling l E(G) ? k,
  k1, , kq-1 such that for each vertex v, the
  sum of the labels of the edges incident with v
  are all equal to the same constant modulo p
  i.e., l(v) c for some fixed c in Zp.
• If k 1, then G is said to be edge-magic.

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Examples 1-Edge-Magic

• The following maximal outerplanar graphs with 6
  vertices are 1-EM.

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Examples 1-Edge-Magic

• In general, G may admits more than one labeling
  to become a k-edge-magic graph with different
  vertex sums.

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Examples k-Edge-Magic

• In general, G may admits more than one labeling
  to become a k-edge-magic graph.

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Necessary Condition

• A necessary condition for a (p,q)-graph G to be
  k-edge-magic is
• Proof
• The sum of all edges is
• Every edge is counted twice in the vertex sums.

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k-Edge-Magic is periodic

• Theorem If a (p,q)-graph G is k-edge-magic then
  it is ptk-edge-magic for all t 0 .

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Cubic Graphs

• Definition 3-regular (p,q)-graph is called a
  cubic graph.
• The relationship between p and q is
• Since q is an integer, p must be even.

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One for All

• Theorem If a cubic graph is k-edge-magic, then
  it is k-edge-magic for all k.
• Proof
• Since every vertex is of degree 3, by adding or
  subtracting 1 to each adjacent edge, the vertex
  sum remains the same.

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Examples Complete Bipartite

• The complete bipartite graph K3,3 is k-edge-magic
  for all k.

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Not Order 4s

• Theorem A cubic graph with order 4s is not
  k-edge-magic for all k.
• Proof
• The number of edges is 6s.
• The necessary condition implies
• It is impossible for all k.

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Möbius Ladders

• The concept of Möbius ladder was introduced by
  Guy and Harry in 1967.
• It is a cubic circulant graph with an even number
  n of vertices, formed from an n-cycle by adding
  edges (called rungs) connecting opposite pairs
  of vertices in the cycle.

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Möbius Ladders

• A möbius ladder ML(2n) with the vertices denoted
  by a1, a2, , a2n. The edges are then a1, a2,
  a2, a3, a2n, a1, a1, an1, a2, an2,
  , an, a2n.

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Labeling Idea

• Splits all edges into two subsets. The first
  subset contains all the edges of C2n. The second
  set contains all middle edges, which forms a
  perfect matching.
• Construct a graceful labeling for the first
  subset, i.e., an arithmetic progression. The rest
  numbers also form an arithmetic progression.

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Labeling Method 1

• Divides the numbers into three subsets 0, 1, 2,
  3, , 2k n-1, 2n, 2n1, 2n2, 2n3, , 2n2k
  3n-1, n, n1, n2, n3, , n2k 2n-1.
• Use the first two subsets to label C2n by the
  following sequence k1, 1, k2, 2, k3, 3, ,
  2k, k, 0, k1, 1, k2, 2, k3, 3, , 2k, k, 0.

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Example of Method 1
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Labeling Method 2

• We label the edges by 1, 1, 2, 2, 3, 3, , k1,
  k1, nk2, k2, nk3, k3, , 2n, 2k1.
• Label the rest numbers, k2, k3, , nk1 to the
  edges in the middle.

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Example of Method 2
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Cylinder Graphs

• Theorem (Lee, Pigg, Cox 1994) The cylinder
  graph CnxP2 is a 1-edge-magic graph if n is odd.

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Cylinder Graphs Examples
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Generalized Petersen Graphs

• The generalized Petersen graphs P(n,k) were first
  studied by Bannai and Coxeter.
• P(n,k) is the graph with vertices vi, ui 0 i
  n-1 and edges vivi1, viui, uiuik, where
  subscripts modulo n and k.
• Theorem The generalized Petersen graph P(n,t) is
  a k-edge-magic graph for all k if n is odd.

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Gen. Petersen Graph Ex.
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Order 6

• Theorem A cubic graph with order 6 is
  k-edge-magic for all k.

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Order 10
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Order 14 Transformation
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Order 14 Transformation
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Conjecture

• Conjecture A cubic graph with order 4s2 is
  k-edge-magic for all k.
• With the previous examples, this is a reasonable
  extension of a conjecture by Lee, Pigg, Cox in
  1994.

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Mod(m)-Edge-Magic Graphs

• A (p,q)-graph G is called Mod(m)-edge-magic (in
  short Mod(m)-EM) if there is an edge labeling l
  E(G) ? 1,2,,q such that for each vertex v, the
  sum of the labels of the edges incident with v
  are all equal to the same constant modulo m
  i.e., l(v) c for some fixed c in Zm.

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Relationship between EM

• Theorem For a graph with order p, if it is
  1-edge-magic, then it is mod(m)-edge-magic for m
  to be a factor of p.
• Proof
• Since m is a factor of p, the constant sum in Zp
  remains constant in Zm.

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Counterexample
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Proof

• Since it is mod(5)-edge-magic, we have relations
  as followings
• a b l m, (1)
• b c k l, (2)
• h i f e, (3)
• From relations (1) and (2), we have
• a k c m. (4)

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Proof (continued)

• Therefore we have g d. Then we have a new
  relation
• h f i e. (5)
• From relations (3) and (5), we have
• i f.
• Then we have
• h e, and
• g j d.

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Proof (continued)

• Without losing generality, we say d 0, e 1
  and f 4.
• From relation (4), we have a 1, k 4, c 2, m
  3 or a 1, k 4, c 3, m 2 or a 2, k
  3, c 1, m 4 or a 2, k 3, c 4, m 1.
• Here, we already run out of 1 and 4 and only 2
  and 3 left in the set.

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Proof (continued)

• For the case a 1, we have b 2 and n 2. If
  forces that o 3 and l 3. But m 3 or 2 cant
  make the sum on v9 equal to 0. This is a
  contradiction.
• With the same argument, we can show that all the
  possibilities cant be true.
• Therefore it is not mod(5)-edge-magic.

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Future Problems

• Do we have just a few counterexamples?
• Any better necessary condition?
• Possible sufficient conditions?