Sachs triangulations, generated by dessins d'enfant, and regular maps
Self-dual planar hypergraphs and exact bond percolation thresholds.
Self-dual regular maps from medial graphs.
Semi-topological classification of line patterns in the plane
Sharp edge-homotopy on spatial graphs.
A sharp-move is known as an unknotting operation for knots. A self sharp-move is a sharp-move on a spatial graph where all strings in the move belong to the same spatial edge. We say that two spatial embeddings of a graph are sharp edge-homotopic if they are transformed into each other by self sharp-moves and ambient isotopies. We investigate how is the sharp edge-homotopy strong and classify all spatial theta curves completely up to sharp edge-homotopy. Moreover we mention a relationship between...
Short cycles of low weight in normal plane maps with minimum degree 5
In this note, precise upper bounds are determined for the minimum degree-sum of the vertices of a 4-cycle and a 5-cycle in a plane triangulation with minimum degree 5: w(C₄) ≤ 25 and w(C₅) ≤ 30. These hold because a normal plane map with minimum degree 5 must contain a 4-star with . These results answer a question posed by Kotzig in 1979 and recent questions of Jendrol’ and Madaras.
Signed ordered knotlike quandle presentations.
Simple and efficient bilayer cross counting.
Skein algebras of the solid torus and symmetric spatial graphs
We use the topological invariant of spatial graphs introduced by S. Yamada to find necessary conditions for a spatial graph to be periodic with a prime period. The proof of the main result is based on computing the Yamada skein algebra of the solid torus and then proving that it injects into the Kauffman bracket skein algebra of the solid torus.
Solution of Heawood's empire problem in the plane.
Some crossing numbers of products of cycles
The exact values of crossing numbers of the Cartesian products of four special graphs of order five with cycles are given and, in addition, all known crossing numbers of Cartesian products of cycles with connected graphs on five vertices are summarized.
Some families of increasing planar maps.
Some Geometric Applications of Dilworth's Theorem.
Some Provably Hard Crossing Number Problems.
Some recent results on domination in graphs
In this paper, we survey some new results in four areas of domination in graphs, namely: (1) the toughness and matching structure of graphs having domination number 3 and which are "critical" in the sense that if one adds any missing edge, the domination number falls to 2; (2) the matching structure of graphs having domination number 3 and which are "critical" in the sense that if one deletes any vertex, the domination number falls to 2; (3) upper bounds...
Some recent results on duality and planarity
Spanning subgraphs of embedded graphs
Special Embeddings of the Complete Graph.
Stability of graphs.
Stacks, queues and tracks: layouts of graph subdivisions.