Computational determination of (3,11) and (4,7) cages

A (k,g)-graph is a k-regular graph of girth g, and a (k,g)-cage is a (k,g)-graph of minimum order. We show that a (3,11)-graph of order 112 found by Balaban in 1973 is minimal and unique. We also show that the order of a (4,7)-cage is 67 and find one example. Finally, we improve the lower bounds on the orders of (3,13)-cages and (3,14)-cages to 202 and 260, respectively. The methods used were a combination of heuristic hill-climbing and an innovative backtrack search

2-limited broadcast domination in grid graphs

We establish upper and lower bounds for the 2-limited broadcast domination number of various grid graphs, in particular the Cartesian product of two paths, a path and a cycle, and two cycles. The upper bounds are derived by explicit constructions. The lower bounds are obtained via linear programming duality by finding lower bounds for the fractional 2-limited multipacking numbers of these graphs

The obstructions for toroidal graphs with no K3,3K_{3,3}'s

Forbidden minors and subdivisions for toroidal graphs are numerous. We consider the toroidal graphs with no $K_{3,3}$-subdivisions that coincide with the toroidal graphs with no $K_{3,3}$-minors. These graphs admit a unique decomposition into planar components and have short lists of obstructions. We provide the complete lists of four forbidden minors and eleven forbidden subdivisions for the toroidal graphs with no $K_{3,3}$'s and prove that the lists are sufficient.Comment: 10 pages, 7 figures, revised version with additional detail

On the edge-reconstruction number of a tree

The edge-reconstruction number ern(G) of a graph G is equal to the minimum number of edge-deleted subgraphs G−e of G which are sufficient to determine G up to isomorphism. Building upon the work of Molina and using results from computer searches by Rivshin and more recent ones which we carried out, we show that, apart from three known exceptions, all bicentroidal trees have edge-reconstruction number equal to 2. We also exhibit the known trees having edge-reconstruction number equal to 3 and we conjecture that the three infinite families of unicentroidal trees which we have found to have edge-reconstruction number equal to 3 are the only ones.peer-reviewe

Simpler Projective Plane Embedding

A projective plane is equivalent to a disk with antipodal points identified. A graph is projective planar if it can be drawn on the projective plane with no crossing edges. A linear time algorithm for projective planar embedding has been described by Mohar. We provide a new approach that takes O(n 2 ) time but is much easier to implement. We programmed a variant of this algorithm and used it to computationally verify the known list of all the projective plane obstructions. Key words: graph algorithms, surface embedding, graph embedding, projective plane, forbidden minor, obstruction 1 Background A graph G consists of a set V of vertices and a set E of edges, each of which is associated with an unordered pair of vertices from V . Throughout this paper, n denotes the number of vertices of a graph, and m is the number of edges. A graph is embeddable on a surface M if it can be drawn on M without crossing edges. Archdeacon's survey [2] provides an excellent introduction to topologica..

Ranking and Unranking Permutations in Linear Time

A ranking function for the permutations on n symbols assigns a unique integer in the range [0; n!\Gamma1] to each of the n! permutations. The corresponding unranking function is the inverse: given an integer between 0 and n!\Gamma1, the value of the function is the permutation having this rank. We present simple ranking and unranking algorithms for permutations that can be computed using O(n) arithmetic operations. Keywords: permutation, ranking, unranking, algorithms for combinatorial problems. 1 Historical Background A permutation of order n is an arrangement of n symbols. For convenience when applying modular arithmetic, this paper considers permutations of f0; 1; 2; :::; n\Gamma1g. The set of all permutations over f0; 1; 2; :::; n\Gamma1g is denoted by S n . There are many applications that call for an array indexed by the permutations in S n [2]. One example is the development of programs that search for Hamilton cycles in particular types of Cayley graphs [10, 11]. To do..