On Mod(k)-Edge-magic Cubic Graphs

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On Mod(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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Edge-Magic Graphs

• Lee, Seah and Tan in 1992 defined that a
  (p,q)-graph G is called edge-magic (in short 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 p i.e., l(v) c
  for some fixed c in Zp.

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Examples Edge-Magic

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

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Examples Edge-Magic

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

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Mod(k)-Edge-Magic Graphs

• Let k 2.
• A (p,q)-graph G is called Mod(k)-edge-magic (in
  short Mod(k)-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 k
  i.e., l(v) c for some fixed c in Zk.

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Examples

• A Mod(k)-EM graph for k 2,3,4,6, but not a
  Mod(5)-EM graph.

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Examples

• The path P4 with 4 vertices is Mod(2)-EM, but not
  Mod(k)-EM for k 3,4.

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Problem

• Chopra, Kwong and Lee in 2006 proposed a problem
  to characterize Mod(2)-EM 3-regular graphs.

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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 Mod(k)-edge-magic
  with vertex sum s (mod k), then it is
  Mod(k)-edge-magic for all other vertex sum s as
  long as gcd(k,3)1.
• Proof
• Since every vertex is of degree 3, by adding or
  subtracting 1 to each adjacent edge, the vertex
  sum increases by 1. Since gcd(k,3)1, it
  generates all.

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Sufficient Condition

• Theorem If a cubic graph G of order p has a
  2-regular subgraph with length ?3p/4? or ?3p/4?,
  then it is Mod(2)-EM.
• Proof
• Note that since G is a cubic graph, p is even.
• We provide two lebelings for each p 4s or 4s2.

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When p 4s

• Two Labelings
• Label the edges of the cycle either by even
  numbers, 2, 4, ..., 6s. The remaining 3s edges
  are labeled by 1, 3, 5, ..., 6s-1.
• Label the edges of the cycle either by odd
  numbers, 1, 3, 5, ..., 6s-1. The remaining 3s
  edges are labeled by even numbers 2, 4, ..., 6s.

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Examples
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When p 4s 2

• Two Labelings
• If G has a cycle with length ?3p/4?. Label the
  edges of the cycle 3s1 by even numbers, 2, 4,
  ..., 6s, 6s2. The remaining 3s2 edges are
  labeled by 1, 3, 5, ..., 6s1,6s3 .
• If G has a cycle with length ?3p/4?. Label the
  edges of the cycle 3s2 by odd numbers, 1, 3, 5,
  ..., 6s3.. The remaining 3s1 edges are labeled
  by even numbers 2, 4, ..., 6s2.

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Examples
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Cylinder Graphs

• Theorem A cylinder graph CnxP2 is Mod(2)-EM if n
  ? 2 (mod 4) for n 3.

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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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Möbius Ladders

• Theorem A Möbius ladder ML(2n) is Mod(2)-EM for
  all n 3.

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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 Mod(2)-EM graph for all k 3 if n is odd.

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Gen. Petersen Graph Ex.
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Turtle Shell Graphs

• Add edges to a cycle C2n with vertices a1, a2, ,
  an, b1, b2, , bn such that a1 is adjacent to b1,
  and ai is adjacent to bn2-i, for i 2, , n.
  The resulting cubic graph is called the turtle
  shell graph of order 2n, denoted by TS(2n).
• Theorem The turtle shell graph TS(2n) is
  Mod(2)-EM for all n 3.

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Turtle Shell Graphs Examples
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Issacs Graphs

• For n gt 3, we denote the graph with vertex set V
  xj, ci,j i 1,2,3, j 1, 2, , n such that
  ci,1, ci,2, , ci,n are three disjoint cycles and
  xj is adjacent to c1,j, c2,j, c3,j.
• We call this graph Issacs graph and denote by
  IS(n).

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Issacs Graphs

• Issacs graphs were first considered by Issacs in
  1975 and investigated in Seymour in 1979.
• They are cubic graphs with perfect matching.
• Theorem The Issacs graph IS(2n) is Mod(2)-EM for
  all n 3.

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Issacs Graphs Examples
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Twisted Cylinder Graphs

• Theorem A twisted cylinder graph TW(n) is
  Mod(2)-EM if n ? 2 (mod 4).
• Proof
• If n ? 2 (mod 4), say n 4k2 then the graph
  TW(n) has order 8k4 and size 6(2k1).
• If it is Mod(2)-EM then it has a 2-regular
  subgraph with length 3(2k1). As TW(n) is
  bipartite, it is impossible.

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Proof (continued)

• Proof
• If n ? 0 (mod 4), say n 4k, then the graph
  TW(n) has order 8k and size 12k.
• We want to show it has a 2-regular subgraph with
  length 6k.
• Label k disjoint 6-cycles a1, a2, a3, a4, b3,
  b2, a5, a6, a7, a8, b7, b6, , a4k-3, a4k-2,
  a4k-1, a4k, b4k-1, b4k-2 by even numbers and all
  the remaining edges by odd numbers.

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Twisted Cylinder Graphs Ex.
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Tutte Graphs

• For any complete binary graph B(2,k), k gt 1, we
  append an edge on the root then hang off of each
  leaf a 2t1-cycle (t gt 2) with t independent
  chords not incident to the leaf.
• We denote this cubic graph by Tutte(B(2,k), t).

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Tutte Graphs

• The cubic graph with longest cycle length 2t1.
• For it is inspired by Tuttes construction of
  Tutte(B(2,1), 2).
• Theorem The Tutte(B(2,k),t) is Mod(2)-EM for all
  k,t 1.

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Tutte Graph Examples
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Sufficient Condition Extended

• Theorem If a cubic graph G of order p has a
  2-regular subgraph with ?3p/4? or ?3p/4? edges,
  then it is Mod(2)-EM.
• Proof
• The same labelings work here.

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Coxeter Graphs

• For n gt 3, we append on each vertex of Cn with a
  star St(3), and then join all the leaves of the
  stars by a cycle C2n. We denote the resulting
  cubic graph by Cox(n).
• Note Cox(n) has 4n vertices.
• Theorem The Coxeter graph Cox(n) is Mod(2)-EM
  for all n 3.

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Coxeter Graph Examples
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Necessary Condition

• Theorem If a cubic graph G of order p is
  Mod(2)-EM, then it has a 2-regular subgraph with
  ?3p/4? or ?3p/4? edges.
• Proof
• As a cubic graph, p must be even.
• Since G has 3p/2 edges, it has either ?3p/4? odd
  and ?3p/4? even edges or ?3p/4? odd and ?3p/4?
  even edges.

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Proof (continued)

• Proof
• Since gcd(2,3)1, if G is Mod(2)-EM with sum 0,
  then it is Mod(2)-EM with sum 1.
• Assume that G is Mod(2)-EM with sum 0.
• With vertex sum equals 0, there are only two
  possible labelings

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Proof (continued)

• Proof

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Proof (continued)

• Proof
• Pick an odd edge. Then there must be another odd
  edge attached to its vertex.
• Keep traveling through odd edges.
• Since there is always another odd edge to travel
  through, you stop only when you reach the initial
  odd edge.

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Classification

• Theorem If a cubic graph G of order p is
  Mod(2)-EM if and only if it has a 2-regular
  subgraph with ?3p/4? or ?3p/4? edges.