With the help of a simple recursive construction we give a computer-assisted proof that the numbe... more With the help of a simple recursive construction we give a computer-assisted proof that the number of graceful labellings of a path of length n grows asymptotically at least as fast as (5=3) n . Results of this type have found surprising applications in topological graph theory.
A Cayley map is a Cayley graph embedded in some orientable surface so that the local rotations at... more A Cayley map is a Cayley graph embedded in some orientable surface so that the local rotations at every vertex are identical. In this series we consider two types of such maps: the balanced and antibalanced
Journal of The London Mathematical Society-second Series, 1999
Abstract It is proved that a graph K has an embedding as a regular map on some closed surface if ... more Abstract It is proved that a graph K has an embedding as a regular map on some closed surface if and only if its automorphism group contains a subgroup G which acts transitively on the oriented edges of K such that the stabiliser G e of every edge e is dihedral of order ...
Skoviera, M. and J. '&Iii, Regular maps from Cayley graphs, Part i: Balanced Cayley maps, Discret... more Skoviera, M. and J. '&Iii, Regular maps from Cayley graphs, Part i: Balanced Cayley maps, Discrete Mathematics IO9 (1992) 265-276.
We define the concepts of an injective colouring and the injective chromatic number of a graph an... more We define the concepts of an injective colouring and the injective chromatic number of a graph and give some upper and lower bounds in general, plus some exact values. We explore in particular the injective chromatic number of the hypercube and put it in the context of previous work on similar concepts, especially the theory of errorcorrecting codes. Finally, we give necessary and sufficient conditions for the injective chromatic number to be equal to the degree for a regular graph.
We show that every 3-connected planar graph has a circular embedding in some nonspherical surface... more We show that every 3-connected planar graph has a circular embedding in some nonspherical surface. More generally, we characterize those planar graphs that have a 2-representative embedding in some nonspherical surface.
Abhandlungen Aus Dem Mathematischen Seminar Der Universitat Hamburg, 1983
The crossing function of a graphG with orientable genusn is defined as a mapping $f:\{ \not 0,1, ... more The crossing function of a graphG with orientable genusn is defined as a mapping $f:\{ \not 0,1, \ldots ,n\} \to \{ \not 0,1,2, \ldots \} $ for whichf(k)=crk (G) the crossing number ofG on the orientable surface of genusk. It is proved that any decreasing convex function $f:\{ \not 0,1, \ldots ,n\} \to \{ \not 0,1,2, \ldots \} $ with $f(n) = \not 0$ is the crossing function of some connected graph.
We show that to each graceful labelling of a path on 2s + 1 vertices, s 2, there corresponds a cu... more We show that to each graceful labelling of a path on 2s + 1 vertices, s 2, there corresponds a current assignment on a 3-valent graph which generates at least 2 2s cyclic oriented triangular embeddings of a complete graph on 12s + 7 vertices. We also show that in this correspondence, two distinct graceful labellings never give isomorphic oriented embeddings. Since the number of graceful labellings of paths on 2s + 1 vertices grows asymptotically at least as fast as (5/3) 2s , this method gives at least 11 s distinct cyclic oriented triangular embedding of a complete graph of order 12s + 7 for all sufficiently large s.
We present a theory of Cayley maps, i.e., embeddings of Cayley graphs into oriented surfaces havi... more We present a theory of Cayley maps, i.e., embeddings of Cayley graphs into oriented surfaces having the same cyclic rotation of generators around each vertex. These maps have often been used to encode symmetric embeddings of graphs. We also present an algebraic theory of Cayley maps and we apply the theory to determine exactly which regular or edge-transitive tilings of the sphere or plane are Cayley maps or Cayley graphs. Our main goal, however, is to provide the general theory so as to make it easier for others to study Cayley maps.
Regular maps whose automorphism groups do not have faithful action on vertices, edges, or faces a... more Regular maps whose automorphism groups do not have faithful action on vertices, edges, or faces are analysed in detail.
With the help of a simple recursive construction we give a computer-assisted proof that the numbe... more With the help of a simple recursive construction we give a computer-assisted proof that the number of graceful labellings of a path of length n grows asymptotically at least as fast as (5=3) n . Results of this type have found surprising applications in topological graph theory.
A Cayley map is a Cayley graph embedded in some orientable surface so that the local rotations at... more A Cayley map is a Cayley graph embedded in some orientable surface so that the local rotations at every vertex are identical. In this series we consider two types of such maps: the balanced and antibalanced
Journal of The London Mathematical Society-second Series, 1999
Abstract It is proved that a graph K has an embedding as a regular map on some closed surface if ... more Abstract It is proved that a graph K has an embedding as a regular map on some closed surface if and only if its automorphism group contains a subgroup G which acts transitively on the oriented edges of K such that the stabiliser G e of every edge e is dihedral of order ...
Skoviera, M. and J. '&Iii, Regular maps from Cayley graphs, Part i: Balanced Cayley maps, Discret... more Skoviera, M. and J. '&Iii, Regular maps from Cayley graphs, Part i: Balanced Cayley maps, Discrete Mathematics IO9 (1992) 265-276.
We define the concepts of an injective colouring and the injective chromatic number of a graph an... more We define the concepts of an injective colouring and the injective chromatic number of a graph and give some upper and lower bounds in general, plus some exact values. We explore in particular the injective chromatic number of the hypercube and put it in the context of previous work on similar concepts, especially the theory of errorcorrecting codes. Finally, we give necessary and sufficient conditions for the injective chromatic number to be equal to the degree for a regular graph.
We show that every 3-connected planar graph has a circular embedding in some nonspherical surface... more We show that every 3-connected planar graph has a circular embedding in some nonspherical surface. More generally, we characterize those planar graphs that have a 2-representative embedding in some nonspherical surface.
Abhandlungen Aus Dem Mathematischen Seminar Der Universitat Hamburg, 1983
The crossing function of a graphG with orientable genusn is defined as a mapping $f:\{ \not 0,1, ... more The crossing function of a graphG with orientable genusn is defined as a mapping $f:\{ \not 0,1, \ldots ,n\} \to \{ \not 0,1,2, \ldots \} $ for whichf(k)=crk (G) the crossing number ofG on the orientable surface of genusk. It is proved that any decreasing convex function $f:\{ \not 0,1, \ldots ,n\} \to \{ \not 0,1,2, \ldots \} $ with $f(n) = \not 0$ is the crossing function of some connected graph.
We show that to each graceful labelling of a path on 2s + 1 vertices, s 2, there corresponds a cu... more We show that to each graceful labelling of a path on 2s + 1 vertices, s 2, there corresponds a current assignment on a 3-valent graph which generates at least 2 2s cyclic oriented triangular embeddings of a complete graph on 12s + 7 vertices. We also show that in this correspondence, two distinct graceful labellings never give isomorphic oriented embeddings. Since the number of graceful labellings of paths on 2s + 1 vertices grows asymptotically at least as fast as (5/3) 2s , this method gives at least 11 s distinct cyclic oriented triangular embedding of a complete graph of order 12s + 7 for all sufficiently large s.
We present a theory of Cayley maps, i.e., embeddings of Cayley graphs into oriented surfaces havi... more We present a theory of Cayley maps, i.e., embeddings of Cayley graphs into oriented surfaces having the same cyclic rotation of generators around each vertex. These maps have often been used to encode symmetric embeddings of graphs. We also present an algebraic theory of Cayley maps and we apply the theory to determine exactly which regular or edge-transitive tilings of the sphere or plane are Cayley maps or Cayley graphs. Our main goal, however, is to provide the general theory so as to make it easier for others to study Cayley maps.
Regular maps whose automorphism groups do not have faithful action on vertices, edges, or faces a... more Regular maps whose automorphism groups do not have faithful action on vertices, edges, or faces are analysed in detail.
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Papers by Jozef Siran