162 research outputs found
Distance coloring of the hexagonal lattice
Motivated by the frequency assignment problem we study the d-distant coloring of the vertices of an infinite plane hexagonal lattice H. Let d be a positive integer. A d-distant coloring of the lattice H is a coloring of the vertices of H such that each pair of vertices distance at most d apart have different colors. The d-distant chromatic number of H, denoted χd(H), is the minimum number of colors needed for a d-distant coloring of H. We give the exact value of χd(H) for any d odd and estimations for any d even
Lightweight paths in graphs
Let k be a positive integer, G be a graph on V(G) containing a path on k vertices, and w be a weight function assigning each vertex v ∈ V(G) a real weight w(y). Upper bounds on the weight [formula] of P are presented, where P is chosen among all paths of G on k vertices with smallest weight
An inequality concerning edges of minor weight in convex 3-polytopes
Let be the number of edges in a convex 3-polytope joining the vertices of degree i with the vertices of degree j. We prove that for every convex 3-polytope there is ; moreover, each coefficient is the best possible. This result brings a final answer to the conjecture raised by B. Grünbaum in 1973
Subgraphs with Restricted Degrees of their Vertices in Large Polyhedral Maps on Compact Two-manifolds
AbstractLet k≥ 2, be an integer and M be a closed two-manifold with Euler characteristic χ(M) ≤ 0. We prove that each polyhedral map G onM , which has at least (8 k2+ 6 k− 6)|χ (M)| vertices, contains a connected subgraph H of order k such that every vertex of this subgraph has, in G, the degree at most 4 k+ 4. Moreover, we show that the bound 4k+ 4 is best possible. Fabrici and Jendrol’ proved that for the sphere this bound is 10 ifk= 2 and 4 k+ 3 if k≥ 3. We also show that the same holds for the projective plane
On Specific Factors in Graphs
It is well known that if G=(V,E) is a connected multigraph and X subset of V is a subset of even order, then G contains a spanning forest H such that each vertex from X has an odd degree in H and all the other vertices have an even degree in H. This spanning forest may have isolated vertices. If this is not allowed in H, then the situation is much more complicated. In this paper, we study this problem and generalize the concepts of even-factors and odd-factors in a unified form
On long cycles through four prescribed vertices of a polyhedral graph
For a 3-connected planar graph G with circumference c ≥ 44 it is proved that G has a cycle of length at least [1/36]c+[20/3] through any four vertices of G
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