3,963 research outputs found
Rainbow domination and related problems on some classes of perfect graphs
Let and let be a graph. A function is a rainbow function if, for every vertex with
, . The rainbow domination number
is the minimum of over all rainbow
functions. We investigate the rainbow domination problem for some classes of
perfect graphs
A successful concept for measuring non-planarity of graphs: the crossing number
AbstractThis paper surveys how the concept of crossing number, which used to be familiar only to a limited group of specialists, emerges as a significant graph parameter. This paper has dual purposes: first, it reviews foundational, historical, and philosophical issues of crossing numbers, second, it shows a new lower bound for crossing numbers. This new lower bound may be helpful in estimating crossing numbers
Intrinsically triple-linked graphs in RP^3
Flapan--Naimi--Pommersheim showed that every spatial embedding of ,
the complete graph on ten vertices, contains a non-split three-component link;
that is, is intrinsically triple-linked in . The work of
Bowlin--Foisy and Flapan--Foisy--Naimi--Pommersheim extended the list of known
intrinsically triple-linked graphs in to include several other
families of graphs. In this paper, we will show that while some of these graphs
can be embedded 3-linklessly in , is intrinsically
triple-linked in .Comment: 23 pages, 6 figures; v2: revised introduction, minor corrections, new
outlines to longer proof
A necessary and sufficient condition for lower bounds on crossing numbers of generalized periodic graphs in an arbitrary surface
Let , and be a graph, a tree and a cycle of order ,
respectively. Let be the complete join of and an empty graph on
vertices. Then the Cartesian product of and can be
obtained by applying zip product on and the graph produced by zip
product repeatedly. Let denote the crossing number of
in an arbitrary surface . If satisfies certain connectivity
condition, then is not less than the sum of the
crossing numbers of its ``subgraphs". In this paper, we introduced a new
concept of generalized periodic graphs, which contains . For a
generalized periodic graph and a function , where is the number
of subgraphs in a decomposition of , we gave a necessary and sufficient
condition for . As an application, we
confirmed a conjecture of Lin et al. on the crossing number of the generalized
Petersen graph in the plane. Based on the condition, algorithms
are constructed to compute lower bounds on the crossing number of generalized
periodic graphs in . In special cases, it is possible to determine
lower bounds on an infinite family of generalized periodic graphs, by
determining a lower bound on the crossing number of a finite generalized
periodic graph.Comment: 26 pages, 20 figure
- …