Download - Cyclic and Bicyclic Decompositions of the Complete Graph into the 4-Cycle with a Pendant Edge
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Cyclic and Bicyclic Decompositions of the Complete Graph into the 4-Cycle
with a Pendant Edge
Daniel “Lupo” Cantrell Gary “Hoser” CokerRobert “Knob” Gardner*
2010 Southeastern MAA ConferenceElon University; Elon, NC
March 26, 2010
*Presenter, East Tennessee State University, Department of Mathematics and Statistics
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Act 1. Decompositions
Steiner Triple Systems
Jakob Steiner
1850s
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Definition. A decomposition of a simple graph H with isomorphic copies of graph G is a set
{ G1, G2, … , Gn}
where Gi G and V(Gi) V(H) for all i, E(Gi) ∩ E(Gj) = Ø if i ≠ j, and
Gi = H.
n
i 1
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Example. There is a decomposition of K5 into 5-cycles.
= U
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Example. There is a decomposition of K7 into 3-cycles: 1 2
5 2
0
16
34
(0,1,3)
(1,2,4)
(2,3,5)(3,4,6)(4,5,0)(5,6,1)(6,0,2)
3
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Definition. A Steiner triple system of order v, STS(v), is a decomposition of the complete graph on v vertices, Kv , into 3-cycles.
Note. We shall restrict today’s presentation to decompositions of complete graphs.
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From the Saint Andrews MacTutor History of Mathematics website.
Jakob Steiner
1796-1863
J. Steiner, Combinatorische Aufgabe, Journal für die Reine und angewandte Mathematik (Crelle’s Journal), 45 (1853), 181-182.
v ≡ 1 or 3 (mod 6) is necessary.
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M. Reiss, Über eine Steinersche combinatorsche Aufgabe welche in 45sten Bande dieses Journals, Seite 181, gestellt worden ist, Journal für die Reine und angewandte Mathematik (Crelle’s Journal), 56 (1859), 326-344.
Theorem. A STS(v) exists if and only if v ≡ 1 or 3 (mod 6).
Note. Sufficiency follows from Reiss.
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Thomas P. Kirkman
1806-1895
From the Saint Andrews MacTutor History of Mathematics website.
T. Kirkman, On a problem in combinations, Cambridge and Dublin Mathematics Journal, 2 (1847), 191-204.
STS(v) iff v ≡ 1 or 3 (mod 6).
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= L
Definition. The 3-cycle with a pendant edge is denoted L and is:
The graph L is sometimes called the lollipop.
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From Bermond’s website: http://www-sop.inria.fr/members/Jean-Claude.Bermond/
Jean-Claude Bermond
J. C. Bermond and J. Schonheim, G-Decompositions of Kn where G has Four Vertices or Less, Discrete Math. 19 (1977), 113-120.
Theorem. An L-decomposition of Kv exists if and only if v ≡ 0 or 1 (mod 8).
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Definition. The 4-cycle with a pendant edge is denoted H and is:
= H
The graph H is sometimes called a kite. We call H, for personal reasons, the Hoser graph.
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From: http://www.d.umn.edu/~dfroncek/alex/ and http://www-direction.inria.fr/international/DS/page_personnelle.html
Alex Rosa
J. C. Bermond, C. Huang, A. Rosa, and D. Sotteau, Decompositions of Complete Graphs into Isomorphic Subgraphs with Five Vertices, Ars Combinatoria 10 (1980), 211-254.
Theorem. An H-decomposition of Kv exists if and only if v ≡ 0 or 1 (mod 5) and v ≥ 11.
Dominique Sotteau
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Act 2. Automorphisms
Cycles and Bicycles
Peltesohn and Gardner
1930s to present
Automorphisms, eh!
Take off!
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Definition. An automorphism of a G-decomposition of H is a permutation of V(H) which fixes the set of copies of G, { G1, G2, … , Gn}.
Recall. A permutation can be classified by its disjoint decomposition into cycles.
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Definition. A permutation of a (finite) set is cyclic if it consists of a single cycle.
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Definition. A permutation of a (finite) set is cyclic if it consists of a single cycle.
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Definition. A permutation of a (finite) set is cyclic if it consists of a single cycle.
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Definition. A permutation of a (finite) set is cyclic if it consists of a single cycle.
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Definition. A permutation of a (finite) set is cyclic if it consists of a single cycle.
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Definition. A permutation of a (finite) set is cyclic if it consists of a single cycle.
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Definition. A permutation of a (finite) set is bicyclic if it consists of two cycles.
MN
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Definition. A permutation of a (finite) set is bicyclic if it consists of two cycles.
MN
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Definition. A permutation of a (finite) set is bicyclic if it consists of two cycles.
MN
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Definition. A permutation of a (finite) set is bicyclic if it consists of two cycles.
MN
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Definition. A permutation of a (finite) set is bicyclic if it consists of two cycles.
MN
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Definition. A permutation of a (finite) set is bicyclic if it consists of two cycles.
MN
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Theorem. A STS(v) admitting a cyclic automorphism exists if and only if
v ≡ 1 or 3 (mod 6), v ≠ 9.
R. Peltesohn, A Solution to Both of Heffter's Difference Problems (in German), Compositio Math. 6 (1939), 251-257.
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Theorem. A bicyclic Steiner Triple System of order v exists if and only if v = M + N ≡ 1 or 3 (mod 6), M ≡ 1 or 3 (mod 6), M ≠ 9 (M > 1), and M | N.
R. Calahan and R. Gardner, Bicyclic Steiner Triple Systems, Discrete Math. 128 (1994), 35-44.
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Theorem. A cyclic L-decomposition of Kv exists if and only if v ≡ 1 (mod 8).
J. C. Bermond and J. Schonheim, G-Decompositions of Kn where G has Four Vertices or Less, Discrete Math. 19 (1977), 113-120.
R. Gardner, Bicyclic Decompositions of Kv into Copies of K3 {e}, Utilitas Mathematica 54 (1998), 51-57.
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Theorem. A bicyclic L-decomposition of Kv exists if and only if (i) N = 2 M and v = M + N ≡ 9 (mod 24), or (ii) M ≡ 1 (mod 8) and N = k M where k ≡ 7 (mod 8).
R. Gardner, Bicyclic Decompositions of Kv into Copies of K3 {e}, Utilitas Mathe-matica 54 (1998), 51-57.
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Act 3. New Results
Hoser Graphs
Cantrell, Coker, Gardner
2010
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Theorem. A cyclic H-decomposition of Kv exists if and only if v ≡ 1 (mod 10).
D. Cantrell, G. D. Coker, and R. Gardner, Cyclic, f-Cyclic, and Bicyclic Decompositions of the Complete Graph into the 4-Cycle with a Pendant Edge, Utilitas Mathematica,to appear.
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A Cyclic H-Decomposition of K11
01
2
3
4
567
8
9
10 (5, 3, 0, 1) - 102 3 1
54
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Theorem. A bicyclic H-decomposition of Kv, exists if and only if (i) M = N ≡ 3 (mod 10), =≥ 13, or(ii) M ≡ 1 (mod 10) and N = k M where k ≡ 9 (mod 10).
D. Cantrell, G. D. Coker, and R. Gardner, Cyclic, f-Cyclic, and Bicyclic Decompositions of the Complete Graph into the 4-Cycle with a Pendant Edge, Utilitas Mathematica,to appear.
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A Bicyclic H-decomposition of K26 With M = N = 13.
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Special Thanks To: Elsinore Beer for the inspiration for this research!
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Good Day, eh!