Lars Døvling Andersen

Generalized latin rectangles II: Embedding

A (p, q, x)-latin rectangle is a rectangular matrix with x symbols in each cell such that each symbol occurs at most p times in each row and at most q times in each column. It is said to be exact if each symbol occurs exactly p times in each row and exactly q times in each column. We give necessary and sufficient conditions for a given (p, q, x)-latin rectangle A to be embeddable in an exact (p, q, x)-latin rectangle E on the same symbols. The same problem is solved with the restriction added, that no symbol occurs more than once in any cell of E, with the restriction that A and E are symmetric squares, and with both these restrictions.

Generalized Latin rectangles I: Construction and decomposition

A (p, q, x)-latin rectangle is a rectangular matrix with x symbols in each cell such that each symbol occurs at most p times in each row and at most q times in each column. We exploit the close correspondence between (p, q, x)-latin rectangles and equitable edge-colourings of certain graphs. This paper contains results on existence and various forms of decomposition of such rectangles. In a sequel paper embedding is considered.

The NP-completeness of finding A-trails in Eulerian graphs and of finding spanning trees in hypergraphs

Abstract Samuel W. Bent and Udi Manber have shown that it is NP-complete to decide whether a simple, 2-connected, plane Eulerian graph has an A-trail, that is, an Eulerian trail in which successive edges are always neighbours in the cyclic, clockwise ordering defined at each vertex of the graph by the given plane representation. We prove, by a different reduction, that the problem remains NP-complete for simple, 3-connected, plane Eulerian graphs for which all face boundaries are 3-cycles or 4-cycles. We then apply this result to show that it is NP-complete to decide whether a linear hypergraph which is regular of degree 3 has a spanning tree.

Embedding Incomplete Latin Squares in Latin Squares Whose Diagonal is Almost Completely Prescribed

We prove that an incomplete Latin square A of side r can be embedded in a Latin square of side n in which all places but one on the diagonal are prescribed if and only if each symbol σ occurs at least 2r − n + f (σ) times in A, f (f) being the number of times a occurs on the prescribed diagonal.

Orthogonal A-Trails of 4-Regular Graphs Embedded in Surfaces of Low Genus

Anton Kotzig has shown that every connected 4-regular plane graph has an A-trail, that is an Euler trail in which any two consecutive edges lie on a common face boundary. We shall characterise the 4-regular plane graphs which contain twoorthogonalA-trails, that is to say two A-trails for which no subtrail of length 2 appears in both A-trails. Our proof gives rise to a polynomial algorithm for deciding if two such A-trails exists. We shall also discuss the corresponding problem for graphs in the projective plane and the torus, and the related problem of deciding when a 2-regular digraph contains two orthogonal Euler trails.

Embedding Latin Squares with Prescribed Diagonal

The paper is concerned with embedding incomplete latin squares in latin squares with prescribed diagonal. A new result about symmetric latin squares is proved and is then applied to improve a theorem of Hilton on idempotent latin squares and answer a question of Lindner about half-idempotent latin squares.

Optimal acyclic edge-coloring of cubic graphs

An acyclic edge-coloring of a graph is a proper edge-coloring such that the subgraph induced by the edges of any two colors is acyclic. The acyclic chromatic index of a graph G is the smallest number of colors in an acyclic edge-coloring of G. We prove that the acyclic chromatic index of a connected cubic graph G is 4, unless G is K4 or K3,3; the acyclic chromatic index of K4 and K3,3 is 5. This result has previously been published by Fiamcik, but his published proof was erroneous.

Removable edges in cyclically 4-edge-connected cubic graphs

We consider various ways of obtaining smaller cyclically 4-edge-connected cubic graphs from a given such graph. In particular, we consider removable edges: an edgee of a cyclically 4-edge-connected cubic graphG is said to be removable ifG′ is also cyclically 4-edge-connected, whereG′ is the cubic graph obtained fromG by deletinge and suppressing the two vertices of degree 2 created by the deletion. We prove that any cyclically 4-edge-connected cubic graphG with at least 12 vertices has at least 1/5(|E(G)| + 12) removable edges, and we characterize the graphs with exactly 1/5(|E(G)| + 12) removable edges.

Algorithms and outerplanar conditions for A-trails in plane Eulerian graphs

The problem of finding A-trails in plane Eulerian graphs is NP-complete even for 3-connected graphs, as shown by the first two authors in an earlier paper. In the present paper, we discuss sufficient conditions for the existence of an A-trail in a 2-connected plane Eulerian graph. They generalize the result that simple 2-connected outerplane Eulerian graphs always have A-trails; we give a polynomial algorithm for finding A-trails in such graphs and show that this algorithm also works for certain multigraphs.

Small Embeddings of Incomplete Idempotent Latin Squares

Publisher Summary An incomplete r x s latin rectangle on t symbols is a partial latin square in which there are no empty cells; it is a latin square if r = s = t. A partial idempotent latin square of side r on r symbols can be embedded in an idempotent latin square on n symbols, for n ≥ 2r + 1. The case is confined when n - r ≥ 5 (of course, we assume that r = s); there are substantial complications when n - r = 3 and also when n - r = 4. It appears that the main obstruction lies in the fact that when the case n - r = 3 is not analyzed properly; a conjecture about edge-coloring cubic bipartite graphs is constructed and, if this conjecture could be solved, one can obtain a complete analysis of the idempotent analogue of Rysers theorem. The idempotent Ryser conditions are not sufficient to ensure that embedding is possible.