3,963 research outputs found

    Rainbow domination and related problems on some classes of perfect graphs

    Full text link
    Let k∈Nk \in \mathbb{N} and let GG be a graph. A function f:V(G)→2[k]f: V(G) \rightarrow 2^{[k]} is a rainbow function if, for every vertex xx with f(x)=∅f(x)=\emptyset, f(N(x))=[k]f(N(x)) =[k]. The rainbow domination number γkr(G)\gamma_{kr}(G) is the minimum of ∑x∈V(G)∣f(x)∣\sum_{x \in V(G)} |f(x)| 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

    Get PDF
    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

    Full text link
    Flapan--Naimi--Pommersheim showed that every spatial embedding of K10K_{10}, the complete graph on ten vertices, contains a non-split three-component link; that is, K10K_{10} is intrinsically triple-linked in R3\mathbb{R}^3. The work of Bowlin--Foisy and Flapan--Foisy--Naimi--Pommersheim extended the list of known intrinsically triple-linked graphs in R3\mathbb{R}^3 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 RP3\mathbb{R}P^3, K10K_{10} is intrinsically triple-linked in RP3\mathbb{R}P^3.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

    Full text link
    Let HH, TT and CnC_n be a graph, a tree and a cycle of order nn, respectively. Let H(i)H^{(i)} be the complete join of HH and an empty graph on ii vertices. Then the Cartesian product H□TH\Box T of HH and TT can be obtained by applying zip product on H(i)H^{(i)} and the graph produced by zip product repeatedly. Let crΣ(H)\textrm{cr}_{\Sigma}(H) denote the crossing number of HH in an arbitrary surface Σ\Sigma. If HH satisfies certain connectivity condition, then crΣ(H□T)\textrm{cr}_{\Sigma}(H\Box T) 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 H□CnH\Box C_n. For a generalized periodic graph GG and a function f(t)f(t), where tt is the number of subgraphs in a decomposition of GG, we gave a necessary and sufficient condition for crΣ(G)≥f(t)\textrm{cr}_{\Sigma}(G)\geq f(t). As an application, we confirmed a conjecture of Lin et al. on the crossing number of the generalized Petersen graph P(4h+2,2h)P(4h+2,2h) in the plane. Based on the condition, algorithms are constructed to compute lower bounds on the crossing number of generalized periodic graphs in Σ\Sigma. 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
    • …
    corecore