Equipartite Graphs Research Articles

Decomposition of the λ -Fold Complete Equipartite Graph into Unicyclic Graphs of Order Five

For a graph G and a subgraph H of a graph G , an H -decomposition of the graph G is a partition of the edge set of G into subsets E i , 1 ≤ i ≤ k , such that each E i induces a graph isomorphic to H . In this paper, it is proved that every simple connected unicyclic graph of order five decomposes the λ -fold complete equipartite graph whenever the necessary conditions are satisfied. This generalizes a result of Huang, *Utilitas Math.* 97 (2015), 109–117.

Decomposition of complete equipartite graphs into paths and cycles of length 2p

Let Pk and Ck respectively denote a path and a cycle on k vertices. In this paper, we give necessary and sufficient conditions for the existence of a complete{P2p+1,C2p}-decomposition of even regular complete equipartite graphs for all prime p.

Vertex‐regular 1‐factorizations in infinite graphs

AbstractThe existence of 1‐factorizations of an infinite complete equipartite graph (with parts of size ) admitting a vertex‐regular automorphism group is known only when and is countable (i.e., for countable complete graphs) and, in addition, is a finitely generated abelian group of order . In this paper, we show that a vertex‐regular 1‐factorization of under the group exists if and only if has a subgroup of order whose index in is . Furthermore, we provide a sufficient condition for an infinite Cayley graph to have a regular 1‐factorization. Finally, we construct 1‐factorizations that contain a given subfactorization, both having a vertex‐regular automorphism group.

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].

On the generalized Oberwolfach problem

The generalized Oberwolfach problem OP t (2 w + 1; N 1 , N 2 , …, N t ; α 1 , α 2 , …, α t ) asks for a factorization of K 2 w + 1 into α i C N i -factors (where a C N i -factor of K 2 w + 1 is a spanning subgraph whose components are cycles of length N i ≥ 3 ) for i = 1, 2, …, t . Necessarily, N = lcm( N 1 , N 2 , …, N t ) is a divisor of 2 w + 1 and w = Σ t i = 1 α i . For t = 1 we have the classic Oberwolfach problem. For t = 2 this is the well-studied Hamilton-Waterloo problem, whereas for t ≥ 3 very little is known. In this paper, we show, among other things, that the above necessary conditions are sufficient whenever 2 w + 1 ≥ ( t + 1) N , α i > 1 for every i ∈ {1, 2, …, t } , and gcd ( N 1 , N 2 , …, N t ) > 1 . We also provide sufficient conditions for the solvability of the generalized Oberwolfach problem over an arbitrary graph and, in particular, the complete equipartite graph.

Simple Minimum (K4 − e)-coverings of Complete Multipartite Graphs

A decomposition of Kn(g) ∪ Γ, the complete n-partite equipartite graph over gn vertices union a graph Γ (called the excess) that is a subgraph of Kn(g), into edge disjoint copies of a graph G is called a simple minimum group divisible covering of type gn with G if Γ contains as few edges as possible. We examine all possible excesses for simple minimum group divisible (K4 − e)-coverings. Necessary and sufficient conditions are established for their existence.

Group divisible ‐packings with any minimum leave

AbstractA decomposition of , the complete n‐partite equipartite graph with a subgraph L (called the leave) removed, into edge disjoint copies of a graph G is called a maximum group divisible packing of with G if L contains as few edges as possible. We examine all possible minimum leaves for maximum group divisible ‐packings. Necessary and sufficient conditions are established for their existences.

On the Hamilton–Waterloo problem with odd cycle lengths

AbstractLet denote the complete graph if v is odd and , the complete graph with the edges of a 1‐factor removed, if v is even. Given nonnegative integers , the Hamilton–Waterloo problem asks for a 2‐factorization of into α ‐factors and β ‐factors, with a ‐factor of being a spanning 2‐regular subgraph whose components are ℓ‐cycles. Clearly, , , and are necessary conditions. In this paper, we extend a previous result by the same authors and show that for any odd the above necessary conditions are sufficient, except possibly when , or when . Note that in the case where v is odd, M and N must be odd. If M and N are odd but v is even, we also show sufficiency but with further possible exceptions. In addition, we provide results on 2‐factorizations of the complete equipartite graph and the lexicographic product of a cycle with the empty graph.

A Generalization of the Hamilton–Waterloo Problem on Complete Equipartite Graphs

AbstractThe Hamilton–Waterloo problem asks for which s and r the complete graph can be decomposed into s copies of a given 2‐factor F1 and r copies of a given 2‐factor F2 (and one copy of a 1‐factor if n is even). In this paper, we generalize the problem to complete equipartite graphs and show that can be decomposed into s copies of a 2‐factor consisting of cycles of length xzm; and r copies of a 2‐factor consisting of cycles of length yzm, whenever m is odd, , , and . We also give some more general constructions where the cycles in a given two factor may have different lengths. We use these constructions to find solutions to the Hamilton–Waterloo problem for complete graphs.

AVD-total-colouring of complete equipartite graphs

An AVD-total-colouring of a simple graph G is a mapping π:V(G)∪E(G)→C, C a set of colours, such that: (i) for each pair of adjacent or incident elements x,y∈V(G)∪E(G), π(x)≠π(y); (ii) for each pair of adjacent vertices x,y∈V(G), sets {π(x)}∪{π(xv):xv∈E(G),v∈V(G)} and {π(y)}∪{π(yv):yv∈E(G),v∈V(G)} are distinct. The AVD-total-chromatic number, χa″(G), is the smallest number of colours for which G admits an AVD-total-colouring. In 2005, Zhang et al. conjectured that χa″(G)≤Δ(G)+3 for any simple graph G. In this article this conjecture is verified for any complete equipartite graph. Moreover, if G is a complete equipartite graph of even order, then it is shown that χa″(G)=Δ(G)+2.

ON DECOMPOSITIONS OF THE COMPLETE EQUIPARTITE GRAPHS Kkm(2t)INTO GREGARIOUS m-CYCLES

For an even integer m at least 4 and any positive integer <TEX>$t$</TEX>, it is shown that the complete equipartite graph <TEX>$K_{km(2t)}$</TEX> can be decomposed into edge-disjoint gregarious m-cycles for any positive integer <TEX>${\kappa}$</TEX> under the condition satisfying <TEX>${\frac{{(m-1)}^2+3}{4m}}$</TEX> < <TEX>${\kappa}$</TEX>. Here it will be called a gregarious cycle if the cycle has at most one vertex from each partite set.

Sunlet Decomposition of Certain Equipartite Graphs

Let L2n stand for the sunlet graph which is a graph that consists of a cycle and an edge terminating in a vertex of degree one attached to each vertex of cycle Cn. The necessary condition for the equipartite graph Kn+I*K̅m to be decomposed into L2n for n≥2 is that the order of L2n must divide n2m2/2, the order of Kn+I*K̅m. In this work, we show that this condition is sufficient for the decomposition. The proofs are constructive using graph theory techniques.

Some gregarious kite decompositions of complete equipartite graphs

A kite decomposition of a complete multipartite graph is said to be gregarious if each kite in the decomposition has its vertices in four different partite sets. In this paper, we give certain necessary and sufficient conditions for the existence of a gregarious kite decomposition of the complete equipartite graph Kn(m) (with n≥4 parts of size m).

($4$, $5)$-CYCLE SYSTEMS OF COMPLETE MULTIPARTITE GRAPHS

In 1981, Alspach conjectured that if $3\leq m_{i}\leq v$, $v$ is odd and $v(v-1)/2= m_{1}+ m_{2}+ \dots + m_{t}$, then the complete graph $K_{v}$ can be decomposed into $t$ cycles of lengths $m_{1}$, $m_{2}$, $\dots$, $m_{t}$ respectively; if $v$ is even, $v(v-2)/2 = m_{1}+ m_{2}+ \dots + m_{t}$, then the complete graph minus a one-factor $K_{v}-F$ can be decomposed into $t$ cycles of lengths $m_{1}$, $m_{2}$, $\dots$, $m_{t}$ respectively. In this paper, we extend the study to the decomposition of the complete equipartite graph $K_{m(n)}$. For $m_{i}\in \{4,5\}$, we prove that the trivial necessary conditions are also sufficient.

Induced Decompositions of Graphs

We consider those graphs G that admit decompositions into copies of a fixed graph F, each copy being an induced subgraph of G. We are interested in finding the extremal graphs with this property, that is, those graphs G on n vertices with the maximum possible number of edges. We discuss the cases where F is a complete equipartite graph, a cycle, a star, or a graph on at most four vertices.

Decomposing complete equipartite graphs into odd square-length cycles: Number of parts even

In this paper we show that the complete equipartite graph with n parts, each of size 2k, decomposes into cycles of length λ2 for any even n≥4, any integer k≥3 and any odd λ such that 3≤λ<2nk and λ divides k. As a corollary, we obtain necessary and sufficient conditions for the decomposition of any complete equipartite graph with an even number of parts into cycles of length p2, where p is prime. In proving our main result, we have also shown the following. Let λ≥3 and n≥4 be odd and even integers, respectively. Then there exists a decomposition of the λ-fold complete equipartite graph with n parts, each of size 2k, into cycles of length λ if and only if λ<2kn. In particular, if we take the complete graph on 2n vertices, remove a 1-factor, then increase the multiplicity of each edge to λ, the resultant graph decomposes into cycles of length λ if and only if λ<2n.

Closed trail decompositions of some classes of regular graphs

If H1,H2,…,Hk are edge-disjoint subgraphs of G such that E(G)=E(H1)∪E(H2)∪⋯∪E(Hk), then we say that H1,H2,…,HkdecomposeG. If each Hi≅H, then we say that H decomposes G and we denote it by H|G. If each Hi is a closed trail, then the decomposition is called a closed trail decomposition of G. In this paper, we consider the decomposition of a complete equipartite graph with multiplicity λ, that is, (Km∘K¯n)(λ), into closed trails of lengths pm1,pm2,…,pmk, where p is an odd prime number or p=4,∑i=1kpmi is equal to the number of edges of the graph and ∘ denotes the wreath product of graphs. A similar result is also proved for (Km×Kn)(λ), where × denotes the tensor product of graphs, if there exists a p-cycle decomposition of the graph. We obtain the following corollary: if k≥3 divides the number of edges of the even regular graph (Km∘K¯n)(λ), then it has a Tk-decomposition, where Tk denotes a closed trail of length k. For m,n≥3, this corollary subsumes the main results of the papers [A. Burgess, M. Šajna, Closed trail decompositions of complete equipartite graphs, J. Combin. Des. 17 (2009) 374–403]; [B.R. Smith, Decomposing complete equipartite graphs into closed trails of length k, Graphs Combin. 26 (2010) 133–140]. We have also partially obtained some results on Tk-decomposition of (Km×Kn)(λ).

Lambda-fold 2-perfect 6-cycle systems in equipartite graphs

A 6-cycle system of a graph G is an edge-disjoint decomposition of G into 6-cycles. Graphs G, for which necessary and sufficient conditions for existence of a 6-cycle system have been found, include complete graphs and complete equipartite graphs. A 6-cycle system of G is said to be 2-perfect if the graph formed by joining all vertices distance 2 apart in the 6-cycles is again an edge-disjoint decomposition of G, this time into 3-cycles, since the distance 2 graph in any 6-cycle is a pair of disjoint 3-cycles.Necessary and sufficient conditions for existence of 2-perfect 6-cycle systems of both complete graphs and complete equipartite graphs are known, and also of λ-fold complete graphs. In this paper, we complete the problem, giving necessary and sufficient conditions for existence of λ-fold 2-perfect 6-cycle systems of complete equipartite graphs.

Distinguishing partitions and asymmetric uniform hypergraphs

A distinguishing partition for an action of a group Γ on a set X is a partition of X that is preserved by no nontrivial element of Γ. As a special case, a distinguishing partition of a graph is a partition of the vertex set that is preserved by no nontrivial automorphism. In this paper we provide a link between distinguishing partitions of complete equipartite graphs and asymmetric uniform hypergraphs. Suppose that m ≥ 1 and n ≥ 2. We show that an asymmetric n -uniform hypergraph with m edges exists if and only if m ≥ f ( n ), where f (2) = f (14) = 6, f (6) = 5, and f ( n )= ⌊ log 2 ( n + 1) ⌋ + 2 otherwise. It follows that a distinguishing partition of K m ( n ) = K n , n , ..., n , or equivalently for the wreath product action S n Wr S m , exists if and only if m ≥ f ( n ).

Degree-equipartite graphs

A graph G of order 2 n is called degree-equipartite if for every n -element set A ⊆ V ( G ) , the degree sequences of the induced subgraphs G [ A ] and G [ V ( G ) ∖ A ] are the same. In this paper, we characterize all degree-equipartite graphs. This answers Problem 1 in the paper by Grünbaum et al. [B. Grünbaum, T. Kaiser, D. Král, and M. Rosenfeld, Equipartite graphs, Israel J. Math. 168 (2008) 431–444].