Complete Symmetric Digraph Research Articles

The λ-Fold Spectrum Problem for the Orientations of the Eight-Cycle

A D-decomposition of a graph (or digraph) G is a partition of the edge set (or arc set) of G into subsets, where each subset induces a copy of the fixed graph D. Graph decomposition finds motivation in numerous practical applications, particularly in the realm of symmetric graphs, where these decompositions illuminate intricate symmetrical patterns within the graph, aiding in various fields such as network design, and combinatorial mathematics, among various others. Of particular interest is the case where G is K*λKv*, the λ-fold complete symmetric digraph on v vertices, that is, the digraph with λ directed edges in each direction between each pair of vertices. For a given digraph D, the set of all values v for which K*λKv* has a D-decomposition is called the λ-fold spectrum of D. An eight-cycle has 22 non-isomorphic orientations. The λ-fold spectrum problem has been solved for one of these oriented cycles. In this paper, we provide a complete solution to the λ-fold spectrum problem for each of the remaining 21 orientations.

Completing the solution of the directed Oberwolfach problem with cycles of equal length

AbstractIn this paper, we give a solution to the last outstanding case of the directed Oberwolfach problem with tables of uniform length. Namely, we address the two‐table case with tables of equal odd length. We prove that the complete symmetric digraph on vertices, denoted , admits a resolvable decomposition into directed cycles of odd length . This completely settles the directed Oberwolfach problem with tables of uniform length.

How to construct the symmetric cycle of length 5 using Haj\'os construction with an adapted Rank Genetic Algorithm

In 2020 Bang-Jensen et. al. generalized the Haj\'os join of two graphs to the class of digraphs and generalized several results for vertex colorings in digraphs. Although, as a consequence of these results, a digraph can be obtained by Haj\'os constructions (directed Haj\'os join and identifying non-adjacent vertices), determining the Haj\'os constructions to obtain the digraph is a complex problem. In particular, Bang-Jensen et al. posed the problem of determining the Haj\'os operations to construct the symmetric 5-cycle from the complete symmetric digraph of order 3 using only Haj\'os constructions. We successfully adapted a rank-based genetic algorithm to solve this problem by the introduction of innovative recombination and mutation operators from graph theory. The Haj\'os Join became the recombination operator and the identification of independent vertices became the mutation operator. In this way, we were able to obtain a sequence of only 16 Haj\'os operations to construct the symmetric cycle of order 5.

The linear $ k $-arboricity of digraphs

<abstract><p>A linear $ k $-diforest is a directed forest in which every connected component is a directed path of length at most $ k $. The linear $ k $-arboricity of a digraph $ D $ is the minimum number of linear $ k $-diforests needed to partition the arcs of $ D $. In this paper, we study the linear $ k $-arboricity for digraphs, and determine the linear $ 3 $-arboricity and linear $ 2 $-arboricity for symmetric complete digraphs and symmetric complete bipartite digraphs.</p></abstract>

On decomposing the complete symmetric digraph into orientations of K_4-e

On decomposing the complete symmetric digraph into orientations of K_4-e

Partitioning edge-coloured complete symmetric digraphs into monochromatic complete subgraphs

Let K→N be the complete symmetric digraph on the positive integers. Answering a question of DeBiasio and McKenney, we construct a 2-colouring of the edges of K→N in which every monochromatic path has density 0.However, if we restrict the length of monochromatic paths in one colour, then no example as above can exist: We show that every (r+1)-edge-coloured complete symmetric digraph (of arbitrary infinite cardinality) containing no directed paths of edge-length ℓi for any colour i≤r can be covered by ∏i∈[r]ℓi pairwise disjoint monochromatic complete symmetric digraphs in colour r+1.Furthermore, we present a stability version for the countable case of the latter result: We prove that the edge-colouring is uniquely determined on a large subgraph, as soon as the upper density of monochromatic paths in colour r+1 is bounded by ∏i∈[r]1ℓi.

Special cases of the quadratic shortest path problem

The quadratic shortest path problem (QSPP) is the problem of finding a path with prespecified start vertex s and end vertex t in a digraph such that the sum of weights of arcs and the sum of interaction costs over all pairs of arcs on the path is minimized. We first consider a variant of the QSPP known as the adjacent QSPP. It was recently proven that the adjacent QSPP on cyclic digraphs cannot be approximated unless hbox {P}=hbox {NP}. Here, we give a simple proof for the same result. We also show that if the quadratic cost matrix is a symmetric weak sum matrix and all s–t paths have the same length, then an optimal solution for the QSPP can be obtained by solving the corresponding instance of the shortest path problem. Similarly, it is shown that the QSPP with a symmetric product cost matrix is solvable in polynomial time. Further, we provide sufficient and necessary conditions for a QSPP instance on a complete symmetric digraph with four vertices to be linearizable. We also characterize linearizable QSPP instances on complete symmetric digraphs with more than four vertices. Finally, we derive an algorithm that examines whether a QSPP instance on the directed grid graph G_{pq} (p,qge 2) is linearizable. The complexity of this algorithm is {mathcal {O}(p^{3}q^{2}+p^{2}q^{3})}.

Linear arboricity of regular digraphs

A linear directed forest is a directed graph in which every component is a directed path. The linear arboricity la(D) of a digraph D is the minimum number of linear directed forests in D whose union covers all arcs of D. For every d-regular digraph D, Nakayama and Peroche conjecture that la(D) = d + 1. In this paper, we consider the linear arboricity for complete symmetric digraphs, regular digraphs with high directed girth and random regular digraphs and we improve some well-known results. Moreover, we propose a more precise conjecture about the linear arboricity for regular digraphs.

On the Directed Oberwolfach Problem with Equal Cycle Lengths

We examine the necessary and sufficient conditions for a complete symmetric digraph $K_n^\ast$ to admit a resolvable decomposition into directed cycles of length $m$. We give a complete solution for even $m$, and a partial solution for odd $m$.

Comparison of relative density of two random geometric digraph families in testing spatial clustering

We compare the performance of relative densities of two parameterized random geometric digraph families called proximity catch digraphs (PCDs) in testing bivariate spatial patterns. These PCD families are proportional edge (PE) and central similarity (CS) PCDs and are defined with proximity regions based on relative positions of data points from two classes. The relative densities of these PCDs were previously used as statistics for testing segregation and association patterns against complete spatial randomness. The relative density of a digraph, D, with n vertices (i.e., with order n) represents the ratio of the number of arcs in D to the number of arcs in the complete symmetric digraph of the same order. When scaled properly, the relative density of a PCD is a U-statistic; hence, it has asymptotic normality by the standard central limit theory of U-statistics. The PE- and CS-PCDs are defined with an expansion parameter that determines the size or measure of the associated proximity regions. In this article, we extend the distribution of the relative density of CS-PCDs for expansion parameter being larger than one, and compare finite sample performance of the tests by Monte Carlo simulations and asymptotic performance by Pitman asymptotic efficiency. We find the optimal expansion parameters of the PCDs for testing each alternative in finite samples and in the limit as the sample size tending to infinity. As a result of our comparisons, we demonstrate that in terms of empirical power (i.e., for finite samples) relative density of CS-PCD has better performance (which occurs for expansion parameter values larger than one) for the segregation alternative, while relative density of PE-PCD has better performance for the association alternative. The methods are also illustrated in a real-life data set from plant ecology.

(P_5)- Factorization of Complete Bipartite Symmetric Digraphs

P2p -factorization of a complete bipartite graph for p, an integer was studied by Wang (1). Further, Beiling (2) extended the work of Wang(1), and studied the P2k -factorization of complete bipartite multigraphs. For even value of k in Pk -factorization the spectrum problem is com- pletely solved (1, 2, 3). However for odd value of k i.e. P3 , P5 and P7 , the path factorization have been studied by a number of researchers (4, 5, 6). Again → P3 -factorization of complete bipartite symmetric digraph was studied by Beiling (7). → P5 -factorization of complete bipartite sym- metric digraph was studied by Rajput and Shukla (8). In the present paper, → P7 -factorization of complete bipartite symmetric digraph has been studied. It is shown that the necessary and sufficient conditions for the existence of → P7 -factorization of complete bipartite symmetric

Cycle decompositions of complete multigraphs

AbstractIn this paper we establish necessary and sufficient conditions for decomposing the complete multigraph λKn into cycles of length λ, and the λ‐fold complete symmetric digraph λK into directed cycles of length λ. As a corollary to these results we obtain necessary and sufficient conditions for decomposing λKn (respectively, λK) into cycles (respectively, directed cycles) of prime length. © 2009 Wiley Periodicals, Inc. J Combin Designs 18: 85–93, 2010

On Mod Difference Labeling of Digraphs

A digraph D = (V,E) is a mod, difference digraph if there exist a positive integer m and a labeling L: V → {1, 2,…, m — 1} such that (x, y) ∊ E if and only if L(y) — L(x) ≡ L(w)(mod m) for some w ∊ V. In this paper we prove that complete symmetric digraphs, unipaths and unicycles are mod difference digraphs.

Irregularity strength of digraphs

It is an elementary exercise to show that any non-trivial simple graph has two vertices with the same degree. This is not the case for digraphs and multigraphs. We consider generating irregular digraphs from arbitrary digraphs by adding multiple arcs. To this end, we define an irregular labeling of a digraph D to be an arc-labeling of the digraph such that the ordered pairs of the sums of the in-labels and out-labels at each vertex are all distinct. We define the strength s → ( D ) of D to be the smallest of the maximum labels used across all irregular labelings. Similar definitions for graphs have been studied extensively and a different formulation of digraph irregularity was given in [H. Hackett, Irregularity strength of graphs and digraphs, Masters Thesis, University of Louisville, 1995]. Here we continue the study of irregular labelings of digraphs. We give a general lower bound on s → ( D ) and determine s → ( D ) exactly for tournaments, directed paths and cycles and the orientation of the path where all vertices have either in-degree 0 or out-degree 0. We also determine the irregularity strength of a union of directed cycles and a union of directed paths, the latter which requires a new result pertaining to finding circuits of given lengths containing prescribed vertices in the complete symmetric digraph with loops.

Directed cyclic Hamiltonian cycle systems of the complete symmetric digraph

In this paper, we prove that directed cyclic Hamiltonian cycle systems of the complete symmetric digraph, K n ∗ , exist if and only if n is odd with n ≠ 15 and n ≠ p α for p an odd prime and α ≥ 2 or n ≡ 2 ( mod 4 ) with n ≠ 2 p α for p an odd prime and α ≥ 1 . We also show that directed cyclic Hamiltonian cycle systems of the complete symmetric digraph minus a set of n / 2 vertex-independent digons, ( K n − I ) ∗ , exist if and only if n ≡ 0 ( mod 4 ) .

Isomorphic factorization, the Kronecker product and the line digraph

In this paper, we investigate isomorphic factorizations of the Kronecker product graphs. Using these relations, it is shown that (1) the Kronecker product of the d-out-regular digraph and the complete symmetric digraph is factorized into the line digraph, (2) the Kronecker product of the Kautz digraph and the de Bruijn digraph is factorized into the Kautz digraph, (3) the Kronecker product of binary generalized de Bruijn digraphs is factorized into the binary generalized de Bruijn digraph.

A note on total colourings of digraphs

In the paper two different arc-colourings and two associated with the total colourings of digraphs are considered. In one of these colourings we show that the problem of calculating the total chromatic index reduces to that of calculating the chromatic number of the underlying graph. In the other colouring we find the total chromatic indices of complete symmetric digraphs and tournaments.

Divisibility conditions in almost Moore digraphs with selfrepeats

Abstract Moore digraph is a digraph with maximum out-degree d, diameter k and order M d , k = 1 + d + … + d k . Moore digraphs exist only in trivial cases if d = 1 (i.e., directed cycle C k ) or k = 1 (i.e., complete symmetric digraph). Almost Moore digraphs are digraphs of order one less than Moore bound. We shall present new properties of almost Moore digraphs with selfrepeats from which we prove nonexistence of almost Moore digraphs for some k and d.

Cycle decompositions IV: complete directed graphs and fixed length directed cycles

We establish necessary and sufficient conditions for decomposing the complete symmetric digraph of order n into directed cycles of length m, where 2⩽ m⩽ n.

Pseudototal colourings of digraphs

In the paper we recall two different colourings of arcs of digraphs and associate them with two special colourings of arcs and vertices of digraphs, called pseudototal colourings of digraphs. We find pseudototal chromatic indices of transitive tournaments and of complete symmetric digraphs.