Decompositions Of Complete Graphs Research Articles

Decomposition of Complete Graphs into Arbitrary Trees

In this paper, we propose a new conjecture that the complete graph $$K_{4m+1}$$ can be decomposed into copies of two arbitrary trees, each of size $$m, m \ge 1$$ . To support this conjecture we prove that the complete graph $$K_{4cm+1}$$ can be decomposed into copies of an arbitrary tree with m edges and copies of the graph H, where H is either a path with m edges or a star with m edges and where c is any positive integer. Further, we discuss related open problems.

Irregularity Strength of Circulant Graphs Using Algorithmic Approach

This paper deals with decomposition of complete graphs on $n$ vertices into circulant graphs with reduced degree $r . They are denoted as $C_{n}(a_{1}, a_{2}, {\dots }, a_{m})$ , where $a_{1}$ to $a_{m}$ are generators. Mathematical labeling for such bigger (higher order and huge size) and complex (strictly regular with so many triangles) graphs is very difficult. That is why after decomposition, an edge irregular $k$ -labeling for these subgraphs is computed with the help of algorithmic approach. Results of $k$ are computed by implementing this iterative algorithm in computer. Using the values of $k$ , an upper bound for edge irregularity strength is suggested for $C_{n}(a_{1}, a_{2}, {\dots }, a_{m})$ that is ${\vert E\vert }/{2}\log _{2} \vert V\vert $ .

Partitioning Transitive Tournaments into Isomorphic Digraphs

In an earlier paper (see Sali and Simonyi Eur. J. Combin. 20, 93–99, 1999) the first two authors have shown that self-complementary graphs can always be oriented in such a way that the union of the oriented version and its isomorphically oriented complement gives a transitive tournament. We investigate the possibilities of generalizing this theorem to decompositions of the complete graph into three or more isomorphic graphs. We find that a complete characterization of when an orientation with similar properties is possible seems elusive. Nevertheless, we give sufficient conditions that generalize the earlier theorem and also imply that decompositions of odd vertex complete graphs to Hamiltonian cycles admit such an orientation. These conditions are further generalized and some necessary conditions are given as well.

Decomposition of complete graphs into connected unicyclic bipartite graphs with eight edges

We prove that each of the 34 non-isomorphic connected unicyclic bipartite graphs with eight edges decomposes the complete graph K n whenever the necesary conditions are satisfied.

Decompositions of some regular graphs into unicyclic graphs of order five

For a graph [Formula: see text] and a subgraph [Formula: see text] of [Formula: see text] an [Formula: see text]-decomposition of [Formula: see text] is a partition of the edge set of [Formula: see text] into subsets [Formula: see text] [Formula: see text] such that each [Formula: see text] induces a graph isomorphic to [Formula: see text] It is proved that the necessary conditions are sufficient for the existence of an [Formula: see text]-decomposition of the graph [Formula: see text] where [Formula: see text] is any simple connected unicyclic graph of order five, × denotes the tensor product of graphs and [Formula: see text] denotes the multiplicity of the edges. In fact, using the above characterization, a necessary and sufficient condition for the graph [Formula: see text] [Formula: see text] and [Formula: see text] to admit an [Formula: see text]-decomposition is obtained. Similar results for the complete graphs and complete multipartite graphs are proved in: [J.-C. Bermond et al. [Formula: see text]-decomposition of [Formula: see text], where [Formula: see text] has four vertices or less, Discrete Math. 19 (1977) 113–120, J.-C. Bermond et al. Decomposition of complete graphs into isomorphic subgraphs with five verices, Ars Combin. 10 (1980) 211–254, M. H. Huang, Decomposing complete equipartite graphs into connected unicyclic graphs of size five, Util. Math. 97 (2015) 109–117].

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

<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>

Uniformly Resolvable Cycle Decompositions with Four Different Factors

In this paper, we consider uniformly resolvable decompositions of complete graph $$K_v$$ (or $$K_v$$ minus a 1-factor I for even v) into cycles. We will focus on the existence of factorizations of $$K_v$$ or $$K_v-I$$ containing up to four non-isomorphic factors. We obtain all possible solutions for uniform factors involving 4, m, 2m and 4m-cycles with a few possible exceptions when m is odd.

Decompositions of Complete Graphs into Bipartite 2-Regular Subgraphs

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 $G$.

New strongly regular decompositions of the complete graphs with prime power vertices

We construct two infinite families of strongly regular decompositions of complete graphs on finite fields, which consist of four negative Latin square type Cayley graphs. Our construction is based on the existence of cyclotomic strongly regular graphs.

Decomposition of complete graphs into (0, 2)-prisms

R. Frucht and J.Gallian (1988) proved that bipartite prisms of order 2n have an α-labeling, thus they decompose the complete graph K 6nx+1 for any positive integer x. We use a technique called the ϱ +-labeling introduced by S. I. El-Zanati, C. Vanden Eynden, and N. Punnim (2001) to show that also some other families of 3-regular bipartite graphs of order 2n called generalized prisms decompose the complete graph K 6nx+1 for any positive integer x.

Decomposition of complete graphs into union of stars

Let S k+1 denote a star with k edges. Tarsi and Yamamoto et al. have characterized the S k+1 –decomposability of K n the complete graph. In this paper we study the edge decomposition of both K n and the complete bipartite graph K m,n into copies of the union of two edge disjoint stars S p+1 and S q+1 where p ≠q and p,q �2 and obtain the necessary and sufficient conditions for the S p+1 ∪Sq+1 -decomposability of K n and K m,n .

Decomposition of complete graphs into small generalized prisms

A prism is the Cartesian product Cm□P2. In other words, it is a graph consisting of two cycles of the same length whose corresponding vertices are joined by additional edges forming a matching. The problem of decomposition of complete graphs into prisms with 12 or 16 vertices was completely solved in [12]. In this paper we completely characterize the complete graphs that are decomposable into 3-regular bipartite graphs of order 12 or 16 that are a simple modification of prisms.

Decomposition of Complete Graphs into Isomorphic Complete Bipartite Graphs

A decomposition of a complete graph into disjoint copies of a complete bipartite graph is called a -design of order n. The existence problem of -designs has been completely solved for the graphs for , for , K2, 3 and K3, 3. In this paper, I prove that for all , if there exists a -design of order N, then there exists a -design of order n for all (mod ) and . Giving necessary direct constructions, I provide an almost complete solution for the existence problem for complete bipartite graphs with fewer than 18 edges, leaving five orders in total unsolved.

Decomposition of Complete Graphs into Cycles and Stars

Let C k denote a cycle of length k and let S k denote a star with k edges. As usual K n denotes the complete graph on n vertices. In this paper we investigate decomposition of K n into C l 's and S k 's, and give some necessary or sufficient conditions for such a decomposition to exist. In particular, we give a complete solution to the problem in the case l = k = 4 as follows: For any nonnegative integers p and q and any positive integer n, there exists a decomposition of K n into p copies of C 4 and q copies of S 4 if and only if $${4(p + q)={n \choose 2}, q\ne 1}$$ if n is odd, and $${q\geq max\{3, \lceil{\frac{n}{4}\rceil}\}}$$ if n is even.

Decomposition of complete graphs into paths and stars

Let P k + 1 denote a path of length k and let S k + 1 denote a star with k edges. As usual K n denotes the complete graph on n vertices. In this paper we investigate the decomposition of K n into paths and stars, and prove the following results. Theorem A. Let p and q be nonnegative integers and let n be a positive integer. There exists a decomposition of K n into p copies of P 4 and q copies of S 4 if and only if n ≥ 6 and 3 ( p + q ) = n 2 . Theorem B. Let p and q be nonnegative integers, let n and k be positive integers such that n ≥ 4 k and k ( p + q ) = n 2 , and let one of the following conditions hold: (1) k is even and p ≥ k 2 , (2) k is odd and p ≥ k . Then there exists a decomposition of K n into p copies of P k + 1 and q copies of S k + 1 .

Decomposition of complete graphs into small graphs

In 1967, A. Rosa proved that if a bipartite graph \(G\) with \(n\) edges has an \(\alpha\)-labeling, then for any positive integer \(p\) the complete graph \(K_{2np+1}\) can be cyclically decomposed into copies of \(G\). This has become a part of graph theory folklore since then. In this note we prove a generalization of this result. We show that every bipartite graph \(H\) which decomposes \(K_k\) and \(K_m\) also decomposes \(K_{km}\).

Decompositions of complete graphs into blown-up cycles [formula omitted

Let C m [ K ¯ 2 ] stand for a cycle C m in which every vertex is replaced by two isolated vertices and every edge by K 2 , 2 . We prove that the complete graph K 8 m k + 1 can be decomposed into graphs isomorphic to C m [ K ¯ 2 ] for any m ≥ 3 , k > 0 . Decompositions of complete graphs into certain collections of even cycles are obtained as a corollary. Also some special cases of Alspach Conjecture are solved in this article. All proofs are constructive and use both graph theory and design theory techniques.

Decompositions of complete graphs into long cycles

The problem of decomposing complete graphs into cycles of arbitrary specified lengths has attracted much attention, but remains largely unsolved. In this paper, the problem is settled in the case where the specified cycle lengths are each more than about half the order of the complete graph. The proof is based on a result that modifies certain existing cycle decompositions to produce new ones in which the lengths of two of the cycles are altered.

Combinatorial designs and the theorem of Weil on multiplicative character sums

In the last years, the theorem of Weil on multiplicative character sums has been very frequently used for getting existence results on combinatorial designs of various kinds. Case by case, the theorem has been applied directly and sometimes this required long and tedious calculations that could be avoided using a result that is a purely algebraic consequence of it. Here this result will be used, in particular, for giving a quick proof of the existence of a ( q , k , λ ) difference family for any admissible prime power q > ( k 2 ) 2 k / g 2 k − 2 where g = gcd ( ( k 2 ) , λ ) , improving in this way the original bound q > ( k 2 ) k 2 − k given by R.M. Wilson [R.M. Wilson, Cyclotomic and difference families in elementary abelian groups, J. Number Theory 4 (1972) 17–47]. More generally, given any simple graph Γ, we prove that there exists an elementary abelian Γ-decomposition of the complete graph K q for any prime power q ≡ 1 (mod 2 e) with q > d 2 e 2 d where d and e are the max–min degree and the number of edges of Γ, respectively. This improves, in some cases enormously, Wilson's bound q > e k 2 − k where k is the number of vertices of Γ (see [R.M. Wilson, Decompositions of complete graphs into subgraphs isomorphic to a given graph, in: C.St.J.A. Nash-Williams, J.H. van Lint (Eds.), Proc. Fifth British Combinatorial Conference. in: Congr. Numer., vol. XV, 1975, pp. 647–659]). The algebraic consequence of the theorem of Weil will be also applied for getting significative existence results on Γ-decompositions of a complete g-partite graph K g × q with q a prime power.

Decomposition of complete graphs into 5‐cubes

AbstractNecessary conditions for the complete graph on n vertices to have a decomposition into 5‐cubes are that 5 divides n − 1 and 80 divides n(n − 1)/2. These are known to be sufficient when n is odd. We prove them also sufficient for n even, thus completing the spectrum problem for the 5‐cube and lending further weight to a long‐standing conjecture of Kotzig. © 2005 Wiley Periodicals, Inc. J Combin Designs 14: 159–166, 2006