Johanan Schönheim

G-decomposition of Kn, where G has four vertices or less

Abstract The complete graph K n , is said to have a G -decomposition if it is the union of edge disjoint subgraphs each isomorphic to G . The set of values of n for which K n has a G -decomposition is determined if G has four vertices or less.

Generalized 1-factorization of trees

Abstract Trees having a 1-factor and trees having a generalized 1-factorization are characterized.

Graph coloring satisfying restraints

Abstract For an integer k ⩾2, a proper k-restraint on a graph G is a function from the vertex set of G to the set of k -colors. A graph G is amenably k-colorable if, for each nonconstant proper k -restraint r on G , there is a k -coloring c of G with c ( v )≠ r ( v ) for each vertex v of G . A graph G is amenable if it is amenably k -colorable and k is the chromatic number of G . For any k ≠3, there are infinitely many amenable k -critical graphs. For k ⩾ 3, we use a construction of B. Toft and amenable graphs to associate a k -colorable graph to any k -colorable finite hypergraph. Some constructions for amenable graphs are given. We also consider a related property—being strongly critical —that is satisfied by many critical graphs, including complete graphs. A strongly critical graph is critical and amenable, but the converse is not always true. The Dirac join operation preserves both amenability and the strongly critical property. In addition, the Hajos construction applied to a single edge in each of two strongly k -critical graphs yields an amenable graph. However, for any k ⩾5, there are amenable k -critical graphs for which the Hajos construction on two copies is not amenable.

On colored designs—II

A colored version of the H-design concept is studied. The problem of whether or not the existence of an H-design for a graph G implies the existence of the corresponding colored designs is solved for small graphs H in the cases of two and three colors.

On colored designs-III: on λ-colored H-designs, H having λ edges

Let H be a graph having λ edges which has no loops and multiple edges. Complete results about the existence of colored designs CH | CKnλ with exactly λ colors, are established for λ ≤ 4 and for arbitrary λ when H is a matching or a star. In all cases when there is a design it can be chosen to be cyclic. Slightly less complete results are obtained when H is a path or a cycle.

Characterization and algorithms of curve map graphs

Abstract A curve map is a planar map obtained by dividing the Euclidean plane into a finite number of regions by a finite set of two-way infinite Jordan curves (every one dividing the plane in two regions) such that no two curves intersect in more than one point. A line map is a curve map obtained by Jordan curves being all straight lines. A graph is called a curve map graph respectively a line map graph if it is the dual of a curve map respectively of a line map. In this paper we give a characterization of the curve map graphs and we describe a polynomial time algorithm for their recognition.

New necessary and sufficient conditions on (ai, mi) in order that x ≡ ai (mod mi) be a covering system

Abstract A covering system is a set of congruences x ≡ ai (mod mi), i = 1, … k, such that every integer satisfies at least one of them. A new necessary and sufficient condition in order that a given set of congruences x ≡ ai (mod mi) be a covering system is established. We show that (4) are such conditions. For exact covering systems they are reduced to (5). The connection of these conditions to known ones such as those [3] based on Bernoulli polynomials and those [8] based on cosets of Zm1 × Zm2 × … × Zmk are studied.

On near subgroups

Abstract If, for every member a of a subset A of elements of an abelian group G, there is an automorphism θa of G such that A + a=θa(A), then A is called a near-subgroup of G. If 0∈A and the order of G is odd, then A is a subgroup of G; otherwise A is not necessarily a coset. However, we show that for a cyclic group of a squarefree order any near-subgroup is a coset. A graph-theoretical motivation is emphasized.

Numerical Decks of Trees

Abstract Let the vertices respectively the edges of a tree T be {v1,v2,…,vn} and {e1,en,…,en-1). The Deck respectively the Edgedeck are then the collections D(T) = {T - vi} and ED(T) = {T-ei} . Each of these decks is redundantly sufficient for reconstructing T. In a numerical approach define the number deck of the tree T the collection ND(T) = {μ1, μ2,…,μ}, where μj = {a1, a2, …,a3} is the collection of integers corresponding to the cardinalities of the connected components of T - vj. Similarly define the edge number deck END(T) and other numerical decks. First the recognition problem is considered, and conditions established for collections of multisets of numbers to be the ND(T) of a tree, the END(T) of a tree, or some other numerical decks considered. It turns out that except for ND(T) the problems are NP-complete. Then, the ND(T) reconstruction problem is considered and a characterization is given.