Un nouveau type de preuve mathématique : le théorème des quatre couleurs. I - Exposé préliminaire
Un nouveau type de preuve mathématiques : II - Le théorème des quatre couleurs
Unavoidable set of face types for planar maps
The type of a face f of a planar map is a sequence of degrees of vertices of f as they are encountered when traversing the boundary of f. A set 𝒯 of face types is found such that in any normal planar map there is a face with type from 𝒯. The set 𝒯 has four infinite series of types as, in a certain sense, the minimum possible number. An analogous result is applied to obtain new upper bounds for the cyclic chromatic number of 3-connected planar maps.
Universal coverings of PL-mainfolds via coloured graphs.
Universal tessellations.
All maps of type (m,n) are covered by a universal map M(m,n) which lies on one of the three simply connected Riemann surfaces; in fact M(m,n) covers all maps of type (r,s) where r|m and s|n. In this paper we construct a tessellation M which is universal for all maps on all surfaces. We also consider the tessellation M(8,3) which covers all triangular maps. This coincides with the well-known Farey tessellation and we find many connections between M(8,3) and M.
Untersuchungen zur Minimalbasis nichtprojektiver Graphen.
Upper embeddable factorizations of graphs
Upward embeddings and orientations of undirected planar graphs.
Upward planarization layout.
User's Guide to Equivariant Methods in Combinatorics
User's guide to equivariant methods in combinatorics. II.
Using graph layout to visualize train interconnection data.
Validity of the weakened Hadwiger hypothesis
Venn diagrams and symmetric chain decompositions in the Boolean lattice.
Venn diagrams with few vertices.
Vertex coloring the square of outerplanar graphs of low degree
Vertex colorings of the square of an outerplanar graph have received a lot of attention recently. In this article we prove that the chromatic number of the square of an outerplanar graph of maximum degree Δ = 6 is 7. The optimal upper bound for the chromatic number of the square of an outerplanar graph of maximum degree Δ ≠ 6 is known. Hence, this mentioned chromatic number of 7 is the last and only unknown upper bound of the chromatic number in terms of Δ.
When a line graph associated to annihilating-ideal graph of a lattice is planar or projective
Let be a finite lattice with a least element 0. is an annihilating-ideal graph of in which the vertex set is the set of all nontrivial ideals of , and two distinct vertices and are adjacent if and only if . We completely characterize all finite lattices whose line graph associated to an annihilating-ideal graph, denoted by , is a planar or projective graph.
WORM Colorings of Planar Graphs
Given three planar graphs F,H, and G, an (F,H)-WORM coloring of G is a vertex coloring such that no subgraph isomorphic to F is rainbow and no subgraph isomorphic to H is monochromatic. If G has at least one (F,H)-WORM coloring, then W−F,H(G) denotes the minimum number of colors in an (F,H)-WORM coloring of G. We show that (a) W−F,H(G) ≤ 2 if |V (F)| ≥ 3 and H contains a cycle, (b) W−F,H(G) ≤ 3 if |V (F)| ≥ 4 and H is a forest with Δ (H) ≥ 3, (c) W−F,H(G) ≤ 4 if |V (F)| ≥ 5 and H is a forest with...
Yamada polynomial and crossing number of spatial graphs.
In this paper we estimate the crossing number of a flat vertex graph in 3-space in terms of the reduced degree of its Yamada polynomial.
Zeros of the Jones polynomial are dense in the complex plane.