scholarly journals A scheduling problem with a simple graphical solution

Author(s):  
A. F. Beecham ◽  
A. C. Hurley

AbstractIt is shown that a problem which arose in the scheduling of two simultaneous competitions between a number of golf clubs may be reduced to that of 4- colouring the edges of a certain bipartite graph which has 4 edges meeting at each vertex. This colouring problem is solved by an analysis in terms of directed cycles, which is simple to carry through in a practical case and is easily extended to the problem with 4 replaced by 2m. The more general colouring problem with 4 replaced by any positive integer is solved by relating it to the marriage problem enunciated by Philip Hall and to the latin multiplication technique of Kaufmann but, in practical applications, this approach involves severe computational difficulties.

Author(s):  
Violeta Cabello ◽  
David Romero ◽  
Ana Musicki ◽  
Ângela Guimarães Pereira ◽  
Baltasar Peñate

AbstractThe literature on the water–energy–food nexus has repeatedly signaled the need for transdisciplinary approaches capable of weaving the plurality of knowledge bodies involved in the governance of different resources. To fill this gap, Quantitative Story-Telling (QST) has been proposed as a science for adaptive governance approach that aims at fostering pluralistic and reflexive research processes to overcome narrow framings of water, energy, and food policies as independent domains. Yet, there are few practical applications of QST and most run on a pan-European scale. In this paper, we apply the theory of QST through a practical case study regarding non-conventional water sources as an innovation for water and agricultural governance in the Canary Islands. We present the methods mixed to mobilize different types of knowledge and analyze interconnections between water, energy, and food supply. First, we map and interview relevant knowledge holders to elicit narratives about the current and future roles of alternative water resources in the arid Canarian context. Second, we run a quantitative diagnosis of nexus interconnections related to the use of these resources for irrigation. This analysis provides feedback to the narratives in terms of constraints and uncertainties that might hamper the expectations posed on this innovation. Thirdly, the mixed analysis is used as fuel for discussion in participatory narrative assessment workshops. Our experimental QST process succeeded in co-creating new knowledge regarding the water–energy–food nexus while addressing some relational and epistemological uncertainties in the development of alternative water resources. Yet, the extent to which mainstream socio-technical imaginaries surrounding this innovation were transformed was rather limited. We conclude that the potential of QST within sustainability place-based research resides on its capacity to: (a) bridge different sources of knowledge, including local knowledge; (b) combine both qualitative and quantitative information regarding the sustainable use of local resources, and (c) co-create narratives on desirable and viable socio-technical pathways. Open questions remain as to how to effectively mobilize radically diverse knowledge systems in complex analytical exercises where everyone feels safe to participate.


2001 ◽  
Vol 63 (3) ◽  
pp. 435-440 ◽  
Author(s):  
Thomas Lam

Dedicated to George Szekeres on his ninetieth birthdayIn this paper we prove that a bipartite graph with parts of sizes M and N, having no cycles of even length less that or equal to 2(2k + 1), where k is a positive integer, has at most (NM)(k+1)/(2k+1) + Dk(N + M) edges, where Dk only depends on k.In particular, we show that when k = 1, D1 = 1 is possible.


2010 ◽  
Vol 97-101 ◽  
pp. 2440-2443
Author(s):  
Shi Jin Wang

Classical flexible job-shop scheduling problem (FJSP) does not consider the transportation time of jobs movement among different machines, which reduces the potential significance of its practical applications. This paper defines an FJSP problem with transportation time incurred by movement of jobs with one Automated Guided Vehicle (AGV) and one Load/Unload (L/U) station. A filtered beam search (FBS) based meta-heuristic algorithm is presented to solve this problem. The detailed procedure of the algorithm is described, and an example is shown to illustrate the algorithm. Finally, preliminary experimental results with comparisons of other dispatching rules demonstrate the feasibility and effectiveness of the proposed algorithm for the FJSP with transportation time.


2014 ◽  
Vol Vol. 16 no. 3 ◽  
Author(s):  
Frederic Havet ◽  
Nagarajan Paramaguru ◽  
Rathinaswamy Sampathkumar

International audience For a connected graph G of order |V(G)| ≥3 and a k-labelling c : E(G) →{1,2,…,k} of the edges of G, the code of a vertex v of G is the ordered k-tuple (ℓ1,ℓ2,…,ℓk), where ℓi is the number of edges incident with v that are labelled i. The k-labelling c is detectable if every two adjacent vertices of G have distinct codes. The minimum positive integer k for which G has a detectable k-labelling is the detection number det(G) of G. In this paper, we show that it is NP-complete to decide if the detection number of a cubic graph is 2. We also show that the detection number of every bipartite graph of minimum degree at least 3 is at most 2. Finally, we give some sufficient condition for a cubic graph to have detection number 3.


2015 ◽  
Vol 58 (2) ◽  
pp. 320-333
Author(s):  
Aurora Llamas ◽  
Josá Martínez–Bernal

AbstractThe cover product of disjoint graphs G and H with fixed vertex covers C(G) and C(H), is the graphwith vertex set V(G) ∪ V(H) and edge setWe describe the graded Betti numbers of GeH in terms of those of. As applications we obtain: (i) For any positive integer k there exists a connected bipartite graph G such that reg R/I(G) = μS(G) + k, where, I(G) denotes the edge ideal of G, reg R/I(G) is the Castelnuovo–Mumford regularity of R/I(G) and μS(G) is the induced or strong matching number of G; (ii)The graded Betti numbers of the complement of a tree depends only upon its number of vertices; (iii)The h-vector of R/I(G e H) is described in terms of the h-vectors of R/I(G) and R/I(H). Furthermore, in a diòerent direction, we give a recursive formula for the graded Betti numbers of chordal bipartite graphs.


1996 ◽  
Vol 160 (1-3) ◽  
pp. 283-289 ◽  
Author(s):  
Hong Wang ◽  
Charles Little ◽  
Kee Teo

1997 ◽  
Vol 13 (3) ◽  
pp. 267-273 ◽  
Author(s):  
Charles Little ◽  
Kee Teo ◽  
Hong Wang

2019 ◽  
Vol 72 (6) ◽  
pp. 1607-1623 ◽  
Author(s):  
Yifan Jing ◽  
Bojan Mohar

AbstractArchdeacon and Grable (1995) proved that the genus of the random graph $G\in {\mathcal{G}}_{n,p}$ is almost surely close to $pn^{2}/12$ if $p=p(n)\geqslant 3(\ln n)^{2}n^{-1/2}$. In this paper we prove an analogous result for random bipartite graphs in ${\mathcal{G}}_{n_{1},n_{2},p}$. If $n_{1}\geqslant n_{2}\gg 1$, phase transitions occur for every positive integer $i$ when $p=\unicode[STIX]{x1D6E9}((n_{1}n_{2})^{-i/(2i+1)})$. A different behaviour is exhibited when one of the bipartite parts has constant size, i.e., $n_{1}\gg 1$ and $n_{2}$ is a constant. In that case, phase transitions occur when $p=\unicode[STIX]{x1D6E9}(n_{1}^{-1/2})$ and when $p=\unicode[STIX]{x1D6E9}(n_{1}^{-1/3})$.


2016 ◽  
Vol 15 (06) ◽  
pp. 1650118
Author(s):  
M. Alinejad ◽  
A. Erfanian

Let [Formula: see text] be a finite group and [Formula: see text] be a subgroup of [Formula: see text]. The non-normal graph of [Formula: see text] in [Formula: see text], denoted by [Formula: see text], is defined as the bipartite graph with two parts [Formula: see text] and [Formula: see text], where [Formula: see text] and [Formula: see text] are the normalizer and the core of [Formula: see text] in [Formula: see text], respectively. Two vertices [Formula: see text] and [Formula: see text] are adjacent if [Formula: see text]. In this paper, we consider vertex and edge connectivity of [Formula: see text]. We show that [Formula: see text] and if [Formula: see text] is a positive integer such that [Formula: see text] and [Formula: see text], then the graph [Formula: see text] has a cycle of length [Formula: see text].


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