Simple Planar Graph Research Articles

List Edge and List Total Colourings of Multigraphs

This paper exploits the remarkable new method of Galvin (J. Combin. Theory Ser. B63(1995), 153–158), who proved that the list edge chromatic numberχ′list(G) of a bipartite multigraphGequals its edge chromatic numberχ′(G). It is now proved here that if every edgee=uwof a bipartite multigraphGis assigned a list of at least max{d(u),d(w)} colours, thenGcan be edge-coloured with each edge receiving a colour from its list. If every edgee=uwin an arbitrary multigraphGis assigned a list of at least max{d(u),d(w)}+⌊12min{d(u),d(w)}⌋ colours, then the same holds; in particular, ifGhas maximum degreeΔ=Δ(G) thenχ′list(G)⩽⌊32Δ⌋. Sufficient conditions are given in terms of the maximum degree and maximum average degree ofGin order thatχ′list(G)=Δandχ″list(G)=Δ+1. Consequences are deduced for planar graphs in terms of their maximum degree and girth, and it is also proved that ifGis a simple planar graph andΔ⩾12 thenχ′list(G)=Δandχ″list(G)=Δ+1.

The maximum sum of degrees above a threshold in planar graphs

We consider the maximum sum of the degrees of vertices with degree at least k in any simple planar graph on n vertices. West and Will solved this for k ⩾ 12 and n ⩾ m k and gave bounds for 6 ⩽ k ⩽ 11. Here we complete the solution by constructing graphs achieving the bounds for all n ⩾ m k when k ⩽ 11.

Decomposition of some planar graphs into trees

We prove that each simple planar graph G whose all faces are quadrilaterals can be decomposed into two disjoint trees T r and T b such that V( T r) = V( G − u) and V( T b) = V( G − v) for any two non-adjacent vertices u and v of G.

Planar graphs with few vertices of small degree

Given n > k > 0, Erdös and Griggs introduced ak(n) = minG | deg(v) < k|, where G runs over all simple planar graphs on n vertices. With many constructions and the Euler formula, we determine ak(n) for all n ⩾ nk: (i) ak(n) = 0, if k < 6; (ii) a6(n) = 4, if n is even; a6(n) = 5, if n is odd; (iii) ak(n) = ⌈ ((k − 6)n + 12)/(k − 3) ⌉, if k = 7,8,9,10; (iv) a11(n) = ⌈ (5n + 18)/8 ⌉; (v)ak(n) = ⌈ ((k − 8)n + 16)/(k − 6)⌉, if k ⩾ 12. The cases k ⩽ 6 follow from results of Grünbaum and Motzkin, while the cases k ⩾ 12 and the lower bound for a11(n) are due to West and Will.

Packings and perfect path double covers of maximal planar graphs

A maximal planar graph is a simple planar graph in which every face is a triangle, and a perfect packing of such a graph by 2-cliques and facial triangles corresponds to a partition of the vertex set into classes, each of which induces either a 2-clique or a facial triangle in the graph. We prove a sufficient condition for a maximal planar graph to have a perfect packing by 2-cliques and facial triangles. This result then leads to a construction of a special type of perfect path double cover of a maximal planar graph.

On maximal and minimal triangular planar graphs: an optimization approach

A triangular planar graph (TPG) is defined to be a connected simple planar graph with every edge being a side of some triangle. Given a fixed number of nodes, a maximal (respectively minimal) TPG is a TPG that has a maximum (respectively minimum) possible number of edges. Here we suggest a novel idea of constructively characterizing a minimal TPG by way of a simple integer optimization formulation, which leads readily to solutions as well as generalizations for planar graphs with edges being sides of polygons. Simple results for the easier problem of maximal TPG are also provided.

Maximal planar graphs of diameter two

AbstractA maximal planar graph is a simple planar graph in which every face is a triangle. We show here that such graphs with maximum degree Δ and diameter two have no more than 3/2Δ + 1 vertices. We also show that there exist maximal planar graphs with diameter two and exactly [3/2Δ + 1] vertices.

Generating all 4‐regular planar graphs from the graph of the octahedron

AbstractIt has been communicated by P. Manca in this journal that all 4‐regular connected planar graphs can be generated from the graph of the octahedron using simple planar graph operations. We point out an error in the generating procedure and correct it by including an additional operation.

Amas dans les graphes planaires

The problem examined in this paper comes from percolation theory. G = ( X, E) is a simple geometric planar graph each vertex of which has a finite degree. We partition X in two subsets X 1, X 2 and we colour in blue each vertex and edge of the subgraph G x generated by X 1 and in red each vertex and edge of G x2 . We obtain blue clusters (resp. red clusters) namely the components of G x 1 (resp. of G x 2). We want to characterize G so that for any such coloration, any finite cluster of one colour is surrounded by a cluster of the other colour. A necessary and sufficient condition is that every component of G is a maximal infinite planar graph and that every vertex x is surrounded by the cycle which connects the vertices adjacent to x.

On the depth of a planar graph

In this paper, we have defined the concept of the depth of a planar graph. We show that, if G is a simple finite planar graph with p vertices and q edges and q > 3( p − 1) − p/2 s−1 , then the depth of G is at least equal to s.

An algorithm for straight-line representation of simple planar graphs

An algorithm is developed for drawing straight-line planar graphs which are isomorphic to a convex polyhedron and simple (i.e. a connected graph with no self-loops or multiple branches). The construction of such graphs is outlined in three stages. Stage 1 determines all the independent cycles of the graph plus one dependent cycle by Goldstein's algorithm. Stage 2 orders the cycles in such a way that they may be drawn in sequence with a given cycle as external. Stage 3 constructs the graph by successively assigning coordinates to the vertices which were placed in order according to the result of Stage 2. An iterative procedure is used for shifting the coordinates to eliminate crossings of edges. Results show that practically all these planar connected graphs up to 22 cycles can be drawn provided that the longest cycle is chosen as the external one. In addition, in the event of failure of the iteration procedure for graphs with a large number of cycles, a heuristic decision involving man-machine interaction is introduced to adjust the partially completed straight-line graph.