Graph equations for line graphs,total graphs,middle graphs and quasi-total graphs

Let G be a graph with vertex-set V(G) and edge-set X(G). Let L(G) and T(G) denote the line graph and total graph of G. The middle graph M(G) of G is an intersection graph Ω(F) on the vertex-set V(G) of any graph G. Let F = V′(G) ∪ X(G) where V′(G) indicates the family of all one-point subsets of the set V(G), then $\text{M(G) = Ω(F)}$.The quasi-total graph P(G) of G is a graph with vertex-set V(G)∪X(G) and two vertices are adjacent if and only if they correspond to two non-adjacent vertices of G or to two adjacent edges of G or to a vertex and an edge incident to it in G.In this paper we solve graph equations $\text{L(G) ? P(H);}\text{L(G)}\text{? P(H); P(G) ? T(H);}\text{P(G)}\text{? T(H); M(G) ? P(H);}\text{M(G)}\text{? P(H)}$.     

Hamiltonian decompositions of complete graphs

It is well known that K_{2n + 1} can be decomposed into n edge-disjoint Hamilton cycles. A novel method for constructing Hamiltonian decompositions of K_{2n + 1} is given and a procedure for obtaining all Hamiltonian decompositions of of K_{2n + 1} is outlined. This method is applied to find a necessary and sufficient condition for a decomposition of the edge set of K_r (r ≤ 2n) into n classes, each class consisting of disjoint paths to be extendible to a Hamiltonian decomposition of K_{2n + 1} so that each of the classes forms part of a Hamilton cycle.     

Commuting decompositions of complete graphs

We say that two graphs G and H with the same vertex set commute if their adjacency matrices commute. In this article, we show that for any natural number r, the complete multigraph K is decomposable into commuting perfect matchings if and only if n is a 2‐power. Also, it is shown that the complete graph K_n is decomposable into commuting Hamilton cycles if and only if n is a prime number. © 2006 Wiley Periodicals, Inc. J Combin Designs     

Isomorphic decomposition of complete graphs into linear forests

The well-known theorem of Kirkman states that every complete graph K_2n of order 2n is 1-factorable or, equivalently, is nK₂-decomposable. This result is generalized to any linear forest of size n without isolated vertices.     

Sharp bounds for decompositions of graphs into complete r-partite subgraphs

If G is a graph on n vertices and r ≥ 2, we let m_r(G) denote the minimum number of complete multipartite subgraphs, with r or fewer parts, needed to partition the edge set, E(G). In determining m_r(G), we may assume that no two vertices of G have the same neighbor set. For such reducedgraphs G, we prove that m_r(G) ≥ log₂ (n + r − 1)/r. Furthermore, for each k ≥ 0 and r ≥ 2, there is a unique reduced graph G = G(r, k) with m_r(G) = k for which equality holds. We conclude with a short proof of the known eigenvalue bound m_r(G) ≥ max{n₊ (G, n₋(G)/(r − 1)}, and show that equality holds if G = G(r, k). © 1996 John Wiley & Sons, Inc.     

Nomadic decompositions of bidirected complete graphs

We use to denote the bidirected complete graph on n vertices. A nomadic Hamiltonian decomposition of is a Hamiltonian decomposition, with the additional property that “nomads” walk along the Hamiltonian cycles (moving one vertex per time step) without colliding. A nomadic near-Hamiltonian decomposition is defined similarly, except that the cycles in the decomposition have length n-1, rather than length n. Bondy asked whether these decompositions of exist for all n. We show that admits a nomadic near-Hamiltonian decomposition when .     

Hamiltonian decompositions of complete regular s-partite graphs

In this paper we give a procedure by which Hamiltonian decompositions of the s-partite graph K_n,n,…,n, where (s − 1)n is even, can be constructed. For 2t⩽s, 1⩽a₁⩽…⩽a_t⩽n, we find conditions which are necessary and sufficient for a decomposition of the edge-set of K_a1a2…,a_t into (s − 1)n/2 classes, each class consisting of disjoint paths, to be extendible to a Hamiltonian decomposition of the complete s-partite graph K_rmn,n,…,n so that each of the classes forms part of a Hamiltonian cycle.     

More on decompositions of edge-colored complete graphs

Let G be a family of graphs whose edges are colored with elements from a set R of r colors. We assume no two vertices of G are joined by more than one edge of color i for any i∈R, for each G∈G. will denote the complete graph with r edges joining any pair of distinct vertices, one of each of the r colors. We describe necessary and asymptotically sufficient conditions on n for the existence of a family D of subgraphs of , each of which is an isomorphic copy of some graph in G, so that each edge of appears in exactly one of the subgraphs in D.     

Closed trail decompositions of complete equipartite graphs

The complete equipartite graph $K_m * {\overline{K_n}}$ has mn vertices partitioned into m parts of size n, with two vertices adjacent if and only if they are in different parts. In this paper, we determine necessary and sufficient conditions for the existence of a decomposition of $K_m * {\overline{K_n}}$ into closed trails of length k. © 2009 Wiley Periodicals, Inc. J Combin Designs 17: 374–403, 2009     

Hamilton decompositions of some line graphs

The main result of this paper completely settles Bermond's conjecture for bipartite graphs of odd degree by proving that if G is a bipartite (2k + 1)-regular graph that is Hamilton decomposable, then the line graph, L(G), of G is also Hamilton decomposable. A similar result is obtained for 5-regular graphs, thus providing further evidence to support Bermond's conjecture. © 1995 John Wiley & Sons, Inc.     

On cyclic decompositions of the complete graph into the 2-regular graphs

The symbol C(m₁ ⁿ ₁m₂ ⁿ ₂...m_s ⁿ _s) denotes a 2-regular graph consisting ofn _i cycles of lengthm _i , i=1, 2,…,s. In this paper, we give some construction methods of cyclic(K _v ,G)-designs, and prove that there exists a cyclic(K _v , G)-design whenG=C((4m ₁) ⁿ ₁(4m ₂) ⁿ ₂...(4m _s ) ⁿ _s andv ≡ 1 (mod 2¦G¦).     

Hamilton cycle decompositions of the tensor products of complete bipartite graphs and complete multipartite graphs

In this paper, it is shown that the tensor product of the complete bipartite graph, K_r,r,r≥2, and the regular complete multipartite graph, , is Hamilton cycle decomposable.     

Resolvable gregarious cycle decompositions of complete equipartite graphs

The complete multipartite graph K_n(m) with n parts of size m is shown to have a decomposition into n-cycles in such a way that each cycle meets each part of K_n(m); that is, each cycle is said to be gregarious. Furthermore, gregarious decompositions are given which are also resolvable.     

On even cycle decompositions of line graphs of cubic graphs

An even cycle decomposition of a graph is a partition of its edges into cycles of even length. In 2012, Markström conjectured that the line graph of every 2-connected cubic graph has an even cycle decomposition and proved this conjecture for cubic graphs with oddness at most 2. However, for 2-connected cubic graphs with oddness 2, Markström only considered these graphs with a chordless 2-factor. (A chordless 2-factor of a graph is a 2-factor consisting of only induced cycles.) In this paper, we first construct an infinite family of 2-connected cubic graphs with oddness 2 and without chordless 2-factors. We then give a complete proof of Markström’s result and further prove this conjecture for cubic graphs with oddness 4.     

Self-orthogonal decompositions of graphs into matchings

Given a simple graph H, a self-orthogonal decomposition (SOD) of H is a collection of subgraphs of H, all isomorphic to some graph G, such that every edge of H occurs in exactly two of the subgraphs and any two of the subgraphs share exactly one edge. Our concept of SOD is a natural generalization of the well-studied orthogonal double covers (ODC) of complete graphs. If for some given G there is an appropriate H, then our goal is to find one with as few vertices as possible. Special attention is paid to the case when G a matching with $n-1$ edges. We conjecture that $v(H)=2n-2$ is best possible if $n\ne 4$ is even and $v(H)=2n$ if n is odd. We present a construction which proves this conjecture for all but 4 of the possible residue classes of n modulo 18.