scholarly journals Decompositions of Complete Graphs into Bipartite 2-Regular Subgraphs

10.37236/4634 ◽  
2016 ◽  
Vol 23 (2) ◽  
Author(s):  
Darryn Bryant ◽  
Andrea Burgess ◽  
Peter Danziger
Keyword(s):  
Regular Graph ◽  
Complete Graph ◽  
Perfect Matching ◽  
Necessary Conditions ◽  
Complete Graphs ◽  
Edge Disjoint

It is shown that if $G$ is any bipartite 2-regular graph of order at most $n/2$ or at least $n-2$, then the obvious necessary conditions are sufficient for the existence of a decomposition of the complete graph of order $n$ into a perfect matching and edge-disjoint copies of $G$.

scholarly journals Decomposition of complete graphs into connected unicyclic graphs with eight edges and pentagon

2019 ◽  
Vol 3 (1) ◽  
pp. 24
Author(s):  
Dalibor Froncek ◽  
O'Neill Kingston
Keyword(s):  
Complete Graph ◽  
Necessary Conditions ◽  
Complete Graphs ◽  
Decomposition Techniques ◽  
Unicyclic Graphs ◽  
Edge Disjoint

<p>A <span class="math"><em>G</em></span>-decomposition of the complete graph <span class="math"><em>K</em><sub><em>n</em></sub></span> is a family of pairwise edge disjoint subgraphs of <span class="math"><em>K</em><sub><em>n</em></sub></span>, all isomorphic to <span class="math"><em>G</em></span>, such that every edge of <span class="math"><em>K</em><sub><em>n</em></sub></span> belongs to exactly one copy of <span class="math"><em>G</em></span>. Using standard decomposition techniques based on <span class="math"><em>ρ</em></span>-labelings, introduced by Rosa in 1967, and their modifications we show that each of the ten non-isomorphic connected unicyclic graphs with eight edges containing the pentagon decomposes the complete graph <span class="math"><em>K</em><sub><em>n</em></sub></span> whenever the necessary conditions are satisfied.</p>

scholarly journals \(C_{6}\)-decompositions of the tensor product of complete graphs

2020 ◽  
Vol 3 (3) ◽  
pp. 62-65
Author(s):  
Abolape Deborah Akwu ◽  
◽  
Opeyemi Oyewumi ◽  
Keyword(s):  
Tensor Product ◽  
Disjoint Union ◽  
Necessary Conditions ◽  
Finite Graph ◽  
Complete Graphs ◽  
Complete Multipartite Graph ◽  
Edge Disjoint ◽  
Multipartite Graph ◽  
Complete Multipartite ◽  
Product Of Graphs

Let \(G\) be a simple and finite graph. A graph is said to be decomposed into subgraphs \(H_1\) and \(H_2\) which is denoted by \(G= H_1 \oplus H_2\), if \(G\) is the edge disjoint union of \(H_1\) and \(H_2\). If \(G= H_1 \oplus H_2 \oplus \cdots \oplus H_k\), where \(H_1\), \(H_2\), ..., \(H_k\) are all isomorphic to \(H\), then \(G\) is said to be \(H\)-decomposable. Furthermore, if \(H\) is a cycle of length \(m\) then we say that \(G\) is \(C_m\)-decomposable and this can be written as \(C_m|G\). Where \( G\times H\) denotes the tensor product of graphs \(G\) and \(H\), in this paper, we prove that the necessary conditions for the existence of \(C_6\)-decomposition of \(K_m \times K_n\) are sufficient. Using these conditions it can be shown that every even regular complete multipartite graph \(G\) is \(C_6\)-decomposable if the number of edges of \(G\) is divisible by \(6\).

scholarly journals Multicolored isomorphic spanning trees in complete graphs

2002 ◽  
Vol Vol. 5 ◽  
Author(s):  
Gregory Constantine
Keyword(s):  
Complete Graph ◽  
Spanning Trees ◽  
Complete Graphs ◽  
Edge Disjoint ◽  
International Audience

International audience Can a complete graph on an even number n (>4) of vertices be properly edge-colored with n-1 colors in such a way that the edges can be partitioned into edge disjoint colorful isomorphic spanning trees? A spanning treee is colorful if all n-1 colors occur among its edges. It is proved that this is possible to accomplish whenever n is a power of two, or five times a power of two.

scholarly journals Extending a perfect matching to a Hamiltonian cycle

2015 ◽  
Vol Vol. 17 no. 1 (Graph Theory) ◽  
Author(s):  
Adel Alahmadi ◽  
Robert E. L. Aldred ◽  
Ahmad Alkenani ◽  
Rola Hijazi ◽  
P. Solé ◽  
...  
Keyword(s):  
Graph Theory ◽  
Regular Graph ◽  
Complete Graph ◽  
Hamiltonian Cycle ◽  
Perfect Matching ◽  
Regular Graphs ◽  
Spanning Subgraph ◽  
International Audience ◽  
Matching Extendability

Graph Theory International audience Ruskey and Savage conjectured that in the d-dimensional hypercube, every matching M can be extended to a Hamiltonian cycle. Fink verified this for every perfect matching M, remarkably even if M contains external edges. We prove that this property also holds for sparse spanning regular subgraphs of the cubes: for every d ≥7 and every k, where 7 ≤k ≤d, the d-dimensional hypercube contains a k-regular spanning subgraph such that every perfect matching (possibly with external edges) can be extended to a Hamiltonian cycle. We do not know if this result can be extended to k=4,5,6. It cannot be extended to k=3. Indeed, there are only three 3-regular graphs such that every perfect matching (possibly with external edges) can be extended to a Hamiltonian cycle, namely the complete graph on 4 vertices, the complete bipartite 3-regular graph on 6 vertices and the 3-cube on 8 vertices. Also, we do not know if there are graphs of girth at least 5 with this matching-extendability property.

P3-factorization of triangulated Cartesian product of complete graphs

2015 ◽  
Vol 07 (01) ◽  
pp. 1450066 ◽  
Author(s):  
A. Tamil Elakkiya ◽  
A. Muthusamy
Keyword(s):  
Necessary Conditions ◽  
Cartesian Product ◽  
Complete Graphs ◽  
Edge Disjoint ◽  
Cartesian Product Of Graphs ◽  
Product Of Graphs

Partition of G into edge-disjoint H-factors is called an H-factorization of G. Shehzad Afzal and Clemens Brand have factorized graphs into factors which are isomorphic to triangulated Cartesian product of two subgraphs. Also they have discussed properties of triangulated Cartesian product of more than two graphs. In this paper, we show that the necessary conditions mn ≡ 0 (mod 3), m, n are odd and 3(mn + m + n - 3) ≡ 0 (mod 8) are sufficient for the existence of a P3-factorization of Km ⧅ Kn, where ⧅ denotes triangulated Cartesian product of graphs, if one of the following holds: (i) m ≡ 9 (mod 12), n = 5, 13 and 17, (ii) m ≡ 9 (mod 12), ns, s > 1, n = 5, 13 and 17, (iii) m ≡ 9 (mod 12), n = psqt for all s, t ≥ 1, where p = 5, 13 and q = 13, 17, p ≠ q, (iv) m ≡ 9 (mod 12), n ≡ 9 (mod 12).

scholarly journals A proof of Ringel’s conjecture

2021 ◽  
Author(s):  
R. Montgomery ◽  
A. Pokrovskiy ◽  
B. Sudakov
Keyword(s):  
Complete Graph ◽  
Complete Graphs ◽  
Edge Disjoint ◽  
Ringel’S Conjecture

AbstractA typical decomposition question asks whether the edges of some graph G can be partitioned into disjoint copies of another graph H. One of the oldest and best known conjectures in this area, posed by Ringel in 1963, concerns the decomposition of complete graphs into edge-disjoint copies of a tree. It says that any tree with n edges packs $$2n+1$$ 2 n + 1 times into the complete graph $$K_{2n+1}$$ K 2 n + 1 . In this paper, we prove this conjecture for large n.

scholarly journals On the Number of Perfect Matchings in Random Lifts

2010 ◽  
Vol 19 (5-6) ◽  
pp. 791-817 ◽  
Author(s):  
CATHERINE GREENHILL ◽  
SVANTE JANSON ◽  
ANDRZEJ RUCIŃSKI
Keyword(s):  
Asymptotic Formula ◽  
Complete Graph ◽  
Pairwise Disjoint ◽  
Perfect Matching ◽  
Perfect Matchings ◽  
Lattice Points ◽  
Laplace’S Method ◽  
Second Moment ◽  
Multiple Dimensions ◽  
Laplace's Method

Let G be a fixed connected multigraph with no loops. A random n-lift of G is obtained by replacing each vertex of G by a set of n vertices (where these sets are pairwise disjoint) and replacing each edge by a randomly chosen perfect matching between the n-sets corresponding to the endpoints of the edge. Let XG be the number of perfect matchings in a random lift of G. We study the distribution of XG in the limit as n tends to infinity, using the small subgraph conditioning method.We present several results including an asymptotic formula for the expectation of XG when G is d-regular, d ≥ 3. The interaction of perfect matchings with short cycles in random lifts of regular multigraphs is also analysed. Partial calculations are performed for the second moment of XG, with full details given for two example multigraphs, including the complete graph K4.To assist in our calculations we provide a theorem for estimating a summation over multiple dimensions using Laplace's method. This result is phrased as a summation over lattice points, and may prove useful in future applications.

Minimal Decompositions of Complete Graphs into Subgraphs with Embeddability Properties

1969 ◽  
Vol 21 ◽  
pp. 992-1000 ◽  
Author(s):  
L. W. Beineke
Keyword(s):  
Projective Plane ◽  
Complete Graph ◽  
Planar Graphs ◽  
Complete Graphs ◽  
Minimum Number ◽  
Double Torus

Although the problem of finding the minimum number of planar graphs into which the complete graph can be decomposed remains partially unsolved, the corresponding problem can be solved for certain other surfaces. For three, the torus, the double-torus, and the projective plane, a single proof will be given to provide the solutions. The same questions will also be answered for bicomplete graphs.

scholarly journals INTRINSICALLY LINKED GRAPHS WITH KNOTTED COMPONENTS

2012 ◽  
Vol 21 (07) ◽  
pp. 1250065 ◽  
Author(s):  
THOMAS FLEMING
Keyword(s):  
Complete Graph ◽  
Cartesian Product ◽  
Complete Graphs ◽  
Linking Number ◽  
A Minor

We construct a graph G such that any embedding of G into R3 contains a nonsplit link of two components, where at least one of the components is a nontrivial knot. Further, for any m < n we produce a graph H so that every embedding of H contains a nonsplit n component link, where at least m of the components are nontrivial knots. We then turn our attention to complete graphs and show that for any given n, every embedding of a large enough complete graph contains a 2-component link whose linking number is a nonzero multiple of n. Finally, we show that if a graph is a Cartesian product of the form G × K2, it is intrinsically linked if and only if G contains one of K5, K3,3 or K4,2 as a minor.

Finding a Longest Alternating Cycle in a 2-edge-coloured Complete Graph is in RP

1996 ◽  
Vol 5 (3) ◽  
pp. 297-306 ◽  
Author(s):  
Rachid Saad
Keyword(s):  
Complete Graph ◽  
Strong Evidence ◽  
General Setting ◽  
Maximum Length ◽  
Sufficient Condition ◽  
Complete Graphs ◽  
Random Polynomial ◽  
Exact Matching ◽  
Matching Problem ◽  
Bipartite Tournament

Jackson [10] gave a polynomial sufficient condition for a bipartite tournament to contain a cycle of a given length. The question arises as to whether deciding on the maximum length of a cycle in a bipartite tournament is polynomial. The problem was considered by Manoussakis [12] in the slightly more general setting of 2-edge coloured complete graphs: is it polynomial to find a longest alternating cycle in such coloured graphs? In this paper, strong evidence is given that such an algorithm exists. In fact, using a reduction to the well known exact matching problem, we prove that the problem is random polynomial.

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