On k-Edge-magic Halin Graphs

1
On k-Edge-magic Halin Graphs
Sin-Min Lee, San Jose State University Hsin-hao
Su, Stonehill College Yung-Chin Wang, Tzu-Hui
Institute of Technology 41th
SICCGTC At Florida Atlantic University March 9,
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-1such 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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Basic Number Theory

• Proposition Let d gcd(a,m).
• ax b has a solution in Zm iff d b.
• Moreover, if d b, then there are exactly d
  solutions in Zm.

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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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Halin Graphs

• Definition Halin graphs are planar connected
  graphs that consist of a tree and a cycle
  connecting the end vertices of the tree.

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Wheels

• For n gt 3, the wheel on n vertices, Wn is a graph
  with n vertices x1, x2,..., xn, x1 having degree
  n-1 and all the other vertices having degree 3.

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Wheels Wn

• of vertices n.
• of edges 2n-2.
• Necessary condition

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Wheels Wns Possible k

• Necessary condition
• Let d gcd(4,n). For t 1,
• If n4t, then d 4.
• If n4t2, then d 2.
• If n4t1 or 4t3, then d 1.
• Possible k
• If n4t, then there is no k.
• If n4t2, then kt2 or 3t3 (mod n).
• If n4t1, then k2t2 (mod n).
• If n4t3, then k2t3 (mod n).

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Wheels not k-EM

• Theorem The Halin graph of Wn for n 4t is not
  k-edge-magic for all k.

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Wheels W5

• Theorem The Halin graph of W5 is k-edge-magic
  for all k 4 (mod 5).

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Wheels W6

• Theorem The Halin graph of W6 is k-edge-magic
  for all k 0,3 (mod 6).

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Wheels W7

• Theorem The Halin graph of W7 is k-edge-magic
  for all k 5 (mod 7).

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Wheels W9

• Theorem The Halin graph of W9 is k-edge-magic
  for all k 6 (mod 9).

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Wheels W2n1

• Theorem The Halin graph of W2n1 is k-edge-magic
  for all k n2 (mod 2n1).

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Double Stars

• Definition The double star D(m,n) is a tree of
  diameter three such that there are m appended
  edges on one ends of P2 and n appended edges on
  another end.

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Double Stars D(m,n)

• of vertices mn2.
• of edges 2(mn)1.
• Necessary condition

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Double Stars Possible k

• Necessary condition
• Let d gcd(6,mn2). Then, d 12.
• Possible k For t 1,
• If mn6t-4, then k2 or 3t1 (mod mn2).
• If mn6t-3, then k2 (mod mn2).
• If mn6t-2, then k2 or 3t1 (mod mn2).
• If mn6t-1, then k2,t2,2t2,3t2,4t2,5t2
  (mod mn2).
• If mn6t, then k2 or 3t3 (mod mn2).
• If mn6t1, then k2 or 2t3 or 4t4 (mod mn2).

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Double Stars D(2,2)

• Theorem The Halin graph of D(2,2), H(D(2,2)), is
  k-edge-magic for all k.

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Double Stars D(2,2)
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Double Stars D(2,3)

• Theorem The Halin graph of D(2,3) is
  k-edge-magic for all k 2 (mod 7).

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Double Stars D(2,4)

• Theorem The Halin graph of D(2,4) is
  k-edge-magic for all k 2,6 (mod 8).

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Double Stars D(3,3)

• Theorem The Halin graph of D(3,3) is
  k-edge-magic for all k 2,6 (mod 8).

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Spiders Sp(2,0,2)

• Theorem The Halin graph of Sp(2,0,2) is
  k-edge-magic for all k 6 (mod 7).

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Spiders Sp(2,0,4)

• Theorem The Halin graph of Sp(2,0,4) is
  k-edge-magic for all k 7 (mod 9).

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Spiders Sp(3,0,3)

• Theorem The Halin graph of Sp(3,0,3) is
  k-edge-magic for all k 7 (mod 9).

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Spiders Sp(2,0,5)

• Theorem The Halin graph of Sp(2,0,5) is
  k-edge-magic for all k 0,5 (mod 10).

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Spiders Sp(3,0,4)

• Theorem The Halin graph of Sp(3,0,4) is
  k-edge-magic for all k 0,5 (mod 10).

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L-Product of Stars with Stars

• Theorem The Halin Graph of St(3)xLSt(2) is
  k-edge-magic for all k 0 (mod 10).

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Spiders Sp(1n,22)

• Theorem The Halin graph of Sp(1n,22) with n
  2i-5 are not k-edge-magic for all k.

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Spiders Sp(11,22)

• Theorem The Halin graph of Sp(11,22) is
  k-edge-magic for all k 1 (mod 6).

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Spiders Sp(12,22)

• Theorem The Halin graph of Sp(12,22) is
  k-edge-magic for all k 6 (mod 7).

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Spiders Sp(14,22)

• Theorem The Halin graph of Sp(14,22) is
  k-edge-magic for all k 7 (mod 9).

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Spiders Sp(15,22)

• Theorem The Halin graph of Sp(15,22) is
  k-edge-magic for all k 0,5 (mod 10).

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Order 8 Not k-EM

• Theorem Among 21 Halin graphs of order 8, the
  following graphs are not k-edge-magic for all k.

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Order 8 2-EM and 6-EM

• Theorem The following Halin graphs of order 8 is
  k-edge-magic for all k 2,6 (mod 8).

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Order 8 3-EM and 7-EM

• Theorem The following Halin graphs of order 8 is
  k-edge-magic for all k 3,7 (mod 8).

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Order 8 3-EM and 7-EM

• Theorem The following Halin graphs of order 8 is
  k-edge-magic for all k 3,7 (mod 8).

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Order 8 3-EM and 7-EM

• Theorem The following Halin graphs of order 8 is
  k-edge-magic for all k 3,7 (mod 8).

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Order 8 3-EM and 7-EM

• Theorem The following Halin graphs of order 8 is
  k-edge-magic for all k 3,7 (mod 8).

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Order 8 3-EM and 7-EM

• Theorem The following Halin graphs of order 8 is
  k-edge-magic for all k 3,7 (mod 8).

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Order 8 3-EM and 7-EM

• Theorem The following Halin graphs of order 8 is
  k-edge-magic for all k 3,7 (mod 8).

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Order 8 7-EM only

• Theorem The following Halin graphs of order 8 is
  k-edge-magic for k 7 (mod 8).

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Order 8 8-EM only

• Theorem The following Halin graphs of order 8 is
  k-edge-magic for k 7 (mod 8).