Edge-disjoint Subgraphs Research Articles

The Ramsey indexes of paths and cycles

Let G be a graph of size m and let c be a red-blue colouring of the edges of G. A Ramsey chain in G with respect to c is a sequence G 1 , G 2 , … , G ℓ of pairwise edge-disjoint subgraphs of G such that each subgraph G i ( 1 ≤ i ≤ ℓ ) is monochromatic of size i and G i is isomorphic to a subgraph of G i + 1 ( 1 ≤ i ≤ ℓ − 1 ). The Ramsey index A R c ( G ) of G with respect to c is the maximum length of a Ramsey chain in G with respect to c. The Ramsey index AR ( G ) of G is the minimum value of A R c ( G ) among all red-blue colourings c of G. Consequently, if G is a graph of size m where ( k + 1 2 ) ≤ m < ( k + 2 2 ) , then AR ( G ) ≤ k . It was proved that if G = m K 2 is a matching of size m or G = K 1 , m is a star of size m, then AR ( G ) = k if and only if ( k + 1 2 ) ≤ m < ( k + 2 2 ) . A question was posed as to whether there are other classes S of graphs with the property that for every sufficiently large integer m, every graph G of size m in S has the property that AR ( G ) = k if and only if ( k + 1 2 ) ≤ m < ( k + 2 2 ) . We show that all paths have this property and, as a consequence, all cycles have this property as well.

Index 3 biembeddings of the complete graphs

We show that the complete graphs on 24s+21 vertices have decompositions into two edge-disjoint subgraphs, each of which triangulates an orientable surface. The special case where the two surfaces are homeomorphic solves a generalized Earth-Moon problem for that surface. Unlike previous constructions, these pairs of triangular embeddings are derived from index 3 current graphs.

Decomposing graphs into interval colorable subgraphs and no-wait multi-stage schedules

A graph G is called interval colorable if it has a proper edge coloring with colors 1 , 2 , 3 , … such that the colors of the edges incident to every vertex of G form an interval of integers. Not all graphs are interval colorable; in fact, quite few families have been proved to admit interval colorings. In this paper we introduce and investigate a new notion, the interval coloring thickness of a graph G , denoted θ int ( G ) , which is the minimum number of interval colorable edge-disjoint subgraphs of G whose union is G . Our investigation is motivated by scheduling problems with compactness requirements, in particular, problems whose solution may consist of several schedules, but where each schedule must not contain any waiting periods or idle times for all involved parties. We first prove that every connected properly 3-edge colorable graph with maximum degree 3 is interval colorable, and using this result, we deduce an upper bound on θ int ( G ) for general graphs G . We demonstrate that this upper bound can be improved in the case when G is bipartite, planar or complete multipartite and consider some applications in timetabling.

The Ascending Ramsey Index of a Graph

Let G be a graph with a given red-blue coloring c of the edges of G. An ascending Ramsey sequence in G with respect to c is a sequence G1, G2, …, Gk of pairwise edge-disjoint subgraphs of G such that each subgraph Gi (1≤i≤k) is monochromatic and Gi is isomorphic to a proper subgraph of Gi+1 (1≤i≤k−1). The ascending Ramsey index ARc(G) of G with respect to c is the maximum length of an ascending Ramsey sequence in G with respect to c. The ascending Ramsey index AR(G) of G is the minimum value of ARc(G) among all red-blue colorings c of G. It is shown that there is a connection between this concept and set partitions. The ascending Ramsey index is investigated for some classes of highly symmetric graphs such as complete graphs, matchings, stars, graphs consisting of a matching and a star, and certain double stars.

Packing and covering immersions in 4-edge-connected graphs

A graph G contains another graph H as an immersion if H can be obtained from a subgraph of G by splitting off edges and removing isolated vertices. In this paper, we prove an edge-variant of the Erdős-Pósa property with respect to the immersion containment in 4-edge-connected graphs. More precisely, we prove that for every graph H, there exists a function f such that for every 4-edge-connected graph G, either G contains k pairwise edge-disjoint subgraphs each containing H as an immersion, or there exists a set of at most f(k) edges of G intersecting all such subgraphs. This theorem is best possible in the sense that the 4-edge-connectivity cannot be replaced by the 3-edge-connectivity.

The edge geodetic self decomposition number of a graph

Let G = (V, E) be a simple connected graph of order p and size q. A decomposition of a graph G is a collection π of edge-disjoint subgraphs G1, G2, … , Gn of G such that every edge of G belongs to exactly one Gi(1 ≤ i ≤ n). The decomposition π = {G1, G2, … , Gn} of a connected graph G is said to be an edge geodetic self decomposition, if ge(Gi) = ge(G) for all i(1 ≤ i ≤ n). The maximum cardinality of π is called the edge geodetic self decomposition number of G and is denoted by πsge(G), where ge(G) is the edge geodetic number of G. Some general properties satisfied by this concept are studied.

Edge geodetic self-decomposition in graphs

Let [Formula: see text] be a simple connected graph of order [Formula: see text] and size [Formula: see text]. A decomposition of a graph [Formula: see text] is a collection of edge-disjoint subgraphs [Formula: see text] of [Formula: see text] such that every edge of [Formula: see text] belongs to exactly one [Formula: see text]. The decomposition [Formula: see text] of a connected graph [Formula: see text] is said to be an edge geodetic self-decomposition if [Formula: see text] for all [Formula: see text]. Some general properties satisfied by this concept are studied.

Structure Fault Tolerance of Recursive Interconnection Networks

Abstract Motivated by effects caused by structure link faults in networks, we study the following graph theoretical problem. Let $T$ be a connected subgraph of a graph $G$ except for $K_{1}$. The $T$-structure edge-connectivity $\lambda (G;T)$ (resp. $T$-substructure edge-connectivity $\lambda ^s(G;T)$) of $G$ is the minimum cardinality of a set of edge-disjoint subgraphs $\mathcal{F}=\{T_{1},T_{2},...,T_{m}\}$ (resp. $\mathcal{F}=\{T_{1}^{^{\prime}},T_{2}^{^{\prime}},...,T_{m}^{^{\prime}}\}$) such that $T_{i}$ is isomorphic to $T$ (resp. $T_{i}^{^{\prime}}$ is a connected subgraph of $T$) for every $1 \le i \le m $, and $E(\mathcal{F})$’s removal leaves the remaining graph disconnected. In this paper, we determine both $\lambda (G;T)$ and $\lambda ^{s}(G;T)$ for $(1)$ the hypercube $Q_{n}$ and $T\in \{K_{1,1},K_{1,2},K_{1,3},P_{4},Q_{1},Q_{2},Q_{3}\}$; $(2)$ the $k$-ary $n$-cube $Q^{k}_{n}$$(k\ge 3)$ and $T\in \{K_{1,1},K_{1,2},K_{1,3},Q^{3}_{1},Q^{4}_{1}\}$; $(3)$ the balanced hypercube $BH_{n}$ and $T\in \{K_{1,1},K_{1,2},BH_{1}\}$. We also extend some results in Lin, C.-K., Zhang, L., Fan, J. and Wang, D. (2016, Structure connectivity and substructure connectivity of hypercubes. Theor. Comput. Sci., 634, 97–107) and Lv, Y., Fan, J., Hsu, D.F. and Lin, C.-K. (2018, Structure connectivity and substructure connectivity of $k$-ary $n$-cubes. Inf. Sci., 433, 115–124).

Computational hardness of spin-glass problems with tile-planted solutions.

We investigate the computational hardness of spin-glass instances on a square lattice, generated via a recently introduced tunable and scalable approach for planting solutions. The method relies on partitioning the problem graph into edge-disjoint subgraphs and planting frustrated, elementary subproblems that share a common local ground state, which guarantees that the ground state of the entire problem is known a priori. Using population annealing Monte Carlo, we compare the typical hardness of problem classes over a large region of the multidimensional tuning parameter space. Our results show that the problems have a wide range of tunable hardness. Moreover, we observe multiple transitions in the hardness phase space, which we further corroborate using simulated annealing and simulated quantum annealing. By investigating thermodynamic properties of these planted systems, we demonstrate that the harder samples undergo magnetic ordering transitions which are also ultimately responsible for the observed hardness transitions on changing the sample composition.

Power of 2 Decomposition of a Complete Tripartite Graph K2,4,M and a Special Butterfly Graph

Let G be a finite, connected simple graph with p vertices and q edges. If G1 , G2 ,…, Gn are connected edge-disjoint subgraphs of G with E(G) = E(G1 )  E(G2 )  …  E(Gn) , then {G1 , G2 , …, Gn} is said to be a decomposition of G. A graph G is said to have Power of 2 Decomposition if G can be decomposed into edge-disjoint subgraphs G G G n  2 4 2 , ,..., such that each G i 2 is connected and ( ) 2 , i E Gi  for 1  i  n. Clearly, 2[2 1] n q . In this paper, we investigate the necessary and sufficient condition for a complete tripartite graph K2,4,m and a Special Butterfly graph           3 2 5 BF 2m 1 to accept Power of 2 Decomposition.

Structure Fault Tolerance of Recursive Interconnection Networks

Abstract Motivated by effects caused by structure link faults in networks, we study the following graph theoretical problem. Let $T$ be a connected subgraph of a graph $G$ except for $K_{1}$. The $T$-structure edge-connectivity $\lambda (G;T)$ (resp. $T$-substructure edge-connectivity $\lambda ^s(G;T)$) of $G$ is the minimum cardinality of a set of edge-disjoint subgraphs $\mathcal{F}=\{T_{1},T_{2},\ldots ,T_{m}\}$ (resp. $\mathcal{F}=\{T_{1}^{^{\prime}},T_{2}^{^{\prime}},\ldots ,T_{m}^{^{\prime}}\}$) such that $T_{i}$ is isomorphic to $T$ (resp. $T_{i}^{^{\prime}}$ is a connected subgraph of $T$) for every $1 \le i \le m$, and $E(\mathcal{F})$’s removal leaves the remaining graph disconnected. In this paper, we determine both $\lambda (G;T)$ and $\lambda ^{s}(G;T)$ for $(1)$ the hypercube $Q_{n}$ and $T\in \{K_{1,1},K_{1,2},K_{1,3},P_{4},Q_{1},Q_{2},Q_{3}\}$; $(2)$ the $k$-ary $n$-cube $Q^{k}_{n}$ $(k\ge 3)$ and $T\in \{K_{1,1},K_{1,2},K_{1,3},Q^{3}_{1},Q^{4}_{1}\}$; $(3)$ the balanced hypercube $BH_{n}$ and $T\in \{K_{1,1},K_{1,2},BH_{1}\}$. We also extend some known results.

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

&lt;p&gt;A &lt;span class="math"&gt;&lt;em&gt;G&lt;/em&gt;&lt;/span&gt;-decomposition of the complete graph &lt;span class="math"&gt;&lt;em&gt;K&lt;/em&gt;&lt;sub&gt;&lt;em&gt;n&lt;/em&gt;&lt;/sub&gt;&lt;/span&gt; is a family of pairwise edge disjoint subgraphs of &lt;span class="math"&gt;&lt;em&gt;K&lt;/em&gt;&lt;sub&gt;&lt;em&gt;n&lt;/em&gt;&lt;/sub&gt;&lt;/span&gt;, all isomorphic to &lt;span class="math"&gt;&lt;em&gt;G&lt;/em&gt;&lt;/span&gt;, such that every edge of &lt;span class="math"&gt;&lt;em&gt;K&lt;/em&gt;&lt;sub&gt;&lt;em&gt;n&lt;/em&gt;&lt;/sub&gt;&lt;/span&gt; belongs to exactly one copy of &lt;span class="math"&gt;&lt;em&gt;G&lt;/em&gt;&lt;/span&gt;. Using standard decomposition techniques based on &lt;span class="math"&gt;&lt;em&gt;ρ&lt;/em&gt;&lt;/span&gt;-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 &lt;span class="math"&gt;&lt;em&gt;K&lt;/em&gt;&lt;sub&gt;&lt;em&gt;n&lt;/em&gt;&lt;/sub&gt;&lt;/span&gt; whenever the necessary conditions are satisfied.&lt;/p&gt;

On the Pendant Number of Some New Graph Classes

Abstract A decomposition of a graph is a collection of its edge disjoint sub-graphs such that their union is . If all the sub-graphs in the decomposition are paths, then it is a path decomposition. In this paper, we discuss the pendant number, the minimum number of end vertices of paths in a path decomposition of a graph. We also determine this parameter for some graph classes. Keywords: Decomposition, path decomposition, pendant number . MSC2010: 05C70, 05C38, 05C40 Cite this Article Jomon K. Sebastian, Joseph Varghese Kureethara, Sudev Naduvath, Charles Dominic. On the Pendant Number of Some New Graph Classes. Research & Reviews: Discrete Mathematical Structures . 2019; 6(1): 15–21p.

Dirac's Condition for Spanning Halin Subgraphs

Let $G$ be an $n$-vertex graph with $n\ge 3$. A classic result of Dirac from 1952 asserts that $G$ is hamiltonian if $\delta(G)\ge n/2$. Dirac's theorem is one of the most influential results in the study of hamiltonicity and by now there are many related known results(see, e.g., [J. A. Bondy, Handbook of Combinatorics, Vol. 1, MIT Press, Cambridge, MA, 1995, pp. 3--110]. A Halin graph is a planar graph consisting of two edge-disjoint subgraphs: a spanning tree of at least four vertices and with no vertex of degree 2, and a cycle induced by the set of the leaves of the spanning tree. Halin graphs possess rich hamiltonicity properties such as being hamiltonian, hamiltonian connected, and almost pancyclic. As a continuous “generalization” of Dirac's theorem, in this paper, we show that there exists a positive integer $n_0$ such that any graph $G$ with $n\ge n_0$ vertices and $\delta(G)\ge (n+1)/2$ contains a spanning and pancyclic Halin subgraph $H$. In addition, for every nonhamiltonian cycle $C$ in $H$, there is a cycle $C'$ longer than $C$ such that $C'$ contains all vertices from $C$ and at most two more vertices not from $C$.

Decomposition of the complete bipartite graph with a 1-factor removed into paths and stars

Let P_k denote a path on k vertices, and let S_k denote a star with k edges. For graphs F, G, and H, a decomposition of F is a set of edge-disjoint subgraphs of F whose union is F. A (G,H)-decomposition of F is a decomposition of F into copies of G and H using at least one of each. In this paper, necessary and sufficient conditions for the existence of the (P_{k+1},S_k)-decomposition of the complete bipartite graph with a 1-factor removed are given.

PENDANT NUMBER OF GRAPHS

A decomposition of a graph G is a collection of its edge disjoint sub-graphs such that their union is G. A path decomposition of a graph is a decomposition of it into paths. In this paper, we define the pendant number Π p as the minimum number of end vertices of paths in a path decomposition of G and determine this parameter for certain fundamental graph classes.

From near to eternity: Spin-glass planting, tiling puzzles, and constraint-satisfaction problems.

We present a methodology for generating Ising Hamiltonians of tunable complexity and with a priori known ground states based on a decomposition of the model graph into edge-disjoint subgraphs. The idea is illustrated with a spin-glass model defined on a cubic lattice, where subproblems, whose couplers are restricted to the two values {-1,+1}, are specified on unit cubes and are parametrized by their local degeneracy. The construction is shown to be equivalent to a type of three-dimensional constraint-satisfaction problem known as the tiling puzzle. By varying the proportions of subproblem types, the Hamiltonian can span a dramatic range of typical computational complexity, from fairly easy to many orders of magnitude more difficult than prototypical bimodal and Gaussian spin glasses in three space dimensions. We corroborate this behavior via experiments with different algorithms and discuss generalizations and extensions to different types of graphs.

K4-expansions have the edge-Erdős-Pósa property

A family F of graphs has the edge-Erdős-Pósa property if there is a function f:N→N such that for every k∈N and every graph G, the following holds: either G contains k edge-disjoint subgraphs contained in F or there is an edge set Y of size at most f(k) such that G−Y does not contain any subgraph in F. We prove that the family of graphs that have K4 as a minor has the edge-Erdős-Pósa property.

Pack Graphs with Subgraphs of Size Three

An $H$-packing $\mathcal{F}$ of a graph $G$ is a set of edge-disjoint subgraphs of $G$ in which each subgraph is isomorphic to $H$. The leave $L$ or the remainder graph $L$ of a packing $\mathcal{F}$ is the subgraph induced by the set of edges of $G$ that does not occur in any subgraph of the packing $\mathcal{F}$. If a leave $L$ contains no edges, or simply $L = \phi$, then $G$ is said to be $H$-decomposable, denoted by $H \mid G$. In this paper, we prove a conjecture made by Chartrand, Saba and Mynhardt [13]: If $G$ is a graph of size $q(G) \equiv 0 \pmod{3}$ and $\delta(G) \geq 2$, then $G$ is $H$-decomposable for some graph $H$ of size $3$.

Near packings of two graphs

A packing of two graph G1 and G2 is a set {H1,H2} such that H1≅G1, H2≅G2, and H1 and H2 are edge disjoint subgraphs of Kn. Let F be a family of graphs. A F-near-packing of G1 and G2 is a generalization of a packing. In a F-near-packing, H1 and H2 may overlap so the subgraph defined by the edges common to both of them is a member of F. In the paper we study three families of graphs (1) Ek –the family of all graphs with at most k edges, (2) Dk –the family of all graphs with maximum degree at most k, and (3) Kk –the family of all graphs having the clique number at most k. By m(n,F) we denote the smallest number m such that there are graphs G1 and G2 both of order n with |E(G1)|+|E(G2)|=m which do not have a F-near-packing. It is well known that m(n,E0)=m(n,D0)=m(n,K1)=3(n−1)2+1 because a E0, D0 or K1-near-packing is just a packing. In this paper we prove that m(n,E1)=m(n,D1)=2n−1 and, for sufficiently large n, m(n,D2)=5(n−1)2+1. We also obtain distinct bounds on m(n,Ek) for k≥2, m(n,D3), and m(n,Kk) for k≥2, under conditions imposed on n and k.