Ascending subgraph decompositions of regular graphs

New constructions of strongly regular Cayley graphs on abelian non p-groups

Let S be a decomposition of a simple 4-regular plane graph into edge-disjoint cycles such that every two adjacent edges on a face belong to different cycles of S. Such graphs, called Grtzsch-Sachs graphs, may be considered as a result of a superposition of simple closed curves in the plane with tangencies disallowed.Koester studied the coloring of Grtzsch-Sachs graphs when all curves are circles.In 1984, he presented the first example of a 4-chromatic edge critical plane graph of order 40 formed by 7 circles.In the present paper, a new 4-chromatic edge critical graph generated by circles in the plane is presented.

R. Haggkvist proved that every 3-regular bipartite graph of order 2n with no component isomorphic to the Heawood graph decomposes the complete bipartite graph K6n,6n. In [1] Cichacz and Froncek established a necessary and sufficient condition for the existence of a factorization of the complete bipartite graph Kn,n into generalized prisms of order 2n. In [2] and [3] Cichacz, Froncek, and Kovar showed decompositions of K3n/2,3n/2 into generalized prisms of order 2n. In this paper we present another class of complete bipartite graphs that is decomposable into prisms.

Short paths in [formula omitted]-regular pairs and small diameter decompositions of dense graphs

A new method is introduced for bounding the separation between the value of $-k$ and the smallest eigenvalue of a non-bipartite $k$-regular graph. The method is based on fractional decompositions of graphs. As a consequence we obtain a very short proof of a generalization and strengthening of a recent result of Qiao, Jing, and Koolen [Electronic J. Combin. 26(2) (2019), #P2.41] about the smallest eigenvalue of non-bipartite distance-regular graphs.

Preface. The Number of Distinct Subwords of Fixed Length in the Morse-Hedlund Sequence S.V. Avgustinovich. Locally Isometric Embeddings of Graphs and the Metric Prolongation Property A.A. Evdokimov. Local Complementations of Simple and Directed Graphs D.G. Fon-Der-Flaass. An Approximation Algorithm for the Travelling Salesman Problem and Its Probabilistic Analysis E.Kh. Gimadi, et al. On Minimum Independent Dominating Sets in Graphs N.I. Glebov, A.V. Kostochka. Regular Partitions and Cuts in Integer Programming A.A. Kolokolov. Complexity of Coverings of Number Sets by Arithmetical Progressions A.D. Korshunov. Circuit Realization of the Sorting Problem E.Sh. Kospanov. A Refinement of the Frank-Sebo-Tardos Theorem and Its Applications A.V. Kostochka. On the Length of the Chinese Postman Tour in Regular Graphs A.V. Kostochka, N. Tulai. An Integer Linear Programming Algorithm Polynomial in the Average Case N.N. Kuzyurin. Projections of the Hypercube on the Line and the Plane A.A. Levin. Canonical Decomposition of Graphs V.V. Lozin. Fault Detection in Parts of the Circuits of Functional Element V.N. Noskov. On the External Stability Number of the Generalized De Bruijn Graphs V. Nyu. On the Lower Bounds for the Complexity of Serial-Parallel Contact Circuits Realizing Linear Boolean Functions K.L. Rychkov. Efficient Scheduling in Open Shops S.V. Sevast'yanov. Nonstrict Vector Summation in Scheduling Problems S.V. Sevast'yanov. Worst-Case Analysis of Some Algorithms for Solving the Subset-Sum Problem Yu.V. Shamardin. On the Depth of Conditional Tests for Controlling 'Negation' Type Faults in Circuits of Functional Gates V.I. Shevchenko. Synthesis of Transitive Order Relations Compatible with the Power ofCriteria L.A. Sholomov. Index.

On the cyclic decomposition of complete graphs into almost-bipartite graphs

A result on decompositions of regular graphs

In 1987, Alavi, Boals, Chartrand, Erdos, and Oellermann conjectured that all graphs have an ascending subgraph decomposition (ASD). Though different classes of graphs have been shown to have an ASD, the conjecture remains open. In this paper we investigate the similar problem for digraphs. In particular, we will show that any orientation of a compete balanced tripartite graph has an ASD.

On the ascending subgraph decomposition problem for bipartite graphs

Let t and q be positive integers that satisfy C ( t + 1 , 2) ≤ q < C ( t + 2 , 2) and let G be a simple and finite graph of size q . G is said to have ascending subgraph decomposition (ASD) if G can be decomposed into t subgraphs H 1 ,H 2 , … ,H t without isolated vertices such that H i is isomorphic to a proper subgraph of H i +1 for 1 ≤ i ≤ t - 1 , where { E ( H 1 ) , … ,E ( H t ) } is a partition of E ( G ) . A graph that admits an ascending subgraph decomposition is called an ASD graph. In this paper, we introduce a new type of magic labeling based on the notion of ASD. Let G be an ASD graph and f : V ( G ) ∪ E ( G ) →{ 1 , 2 , … , | V ( G ) | + | E ( G ) |} be a bijection. The weight of a subgraph H i (1 ≤ i ≤ n ) is w ( H i ) = ∑ v ∈ V ( H i ) f ( v ) + ∑ e ∈ E ( H i ) f ( e ) . If the weight of each ascending subgraph is constant, say w ( H i ) = k , ∀ 1 ≤ i ≤ t , then f is called an ASD-magic labeling of G and G is called an ASD-magic graph. We present general properties of ASD-magic graphs and characterize certain classes of them.