Decomposing Graphs

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Decomposing Graphs into Specified Subgraphs
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Definitions

• Let G be an undirected simple graph and H be a
  subgraph of G.
• G is H-decomposable, denoted by HG, if its edge
  set E(G) can be decomposed into subgraphs such
  that each subgraph is isomorphic to H.
• G has an H-decomposition if G is H-decomposable.

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Motivation
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• It is interesting to us for studying the
  H-decompositions of a graph G with H of size at
  most three.

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Theorem 2.
(Chen and Huang)
Suppose G is a graph of even size and different
from K3?K2 . Then G is M2-decomposable if and
only if q(G) ?2?(G).
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• The Conjecture 1 is not true in general.

Counterexamples
(C. Sunil Kumar)
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• Theorem 3.

(C. Sunil Kumar)
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Main results
2. H-decompositions of graphs with H of size
three.
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are mutually disjoint.
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Theorem 4.
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Proof.
Induction on
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Theorem 5.
The Conjecture 2 is affirmative.
Proof.
(1) If q(G) 3, then GG. (2) If G K4 or
K1,1,3c1, then P4G. (3) Otherwise, by Theorem
4, we have (P3?P2)G.
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2. H-decompositions of graphs with H of size
three.
2.1 H-decompositions of complete multipartite
graphs.
2.2 H-decompositions of cubic graphs.
2.3 H-decompositions of hypercubes.
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2.1 H-decompositions of complete multipartite
graphs.
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• The K3-decomposability of complete multipartite
  graphs is still widely open.

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2.2 H-decompositions of cubic graphs.
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Theorem 6.
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Conjecture 1.
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2.3 H-decompositions of hypercubes.
, is defined recursively by
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(5)
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Theorem 7.
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Theorem 8.
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(Vizing)
(De Werra)
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Corollary 11.
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Theorem 12.
Suppose G is a graph of size q(G)
2?(G). Then ?(G) ?(G) 1 if and only if G
K3?K2.
Theorem 13.
Suppose G is a graph of size q(G) 3?(G)
and ?(G) ?5. Then ?(G) ?(G).
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Proof.
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Example
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Conjecture 00.
(Chen, Huang and Tsai)
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Proof.
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(a) connected.
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(b) disconnected
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