scholarly journals Connected, Bounded Degree, Triangle Avoidance Games

10.37236/680 ◽  
2011 ◽  
Vol 18 (1) ◽  
Author(s):  
Nishali Mehta ◽  
Ákos Seress
Keyword(s):  
Winning Strategy ◽  
Simple Graph ◽  
Bounded Degree ◽  
Empty Graph ◽  
Graph Game

We consider variants of the triangle-avoidance game first defined by Harary and rediscovered by Hajnal a few years later. A graph game begins with two players and an empty graph on $n$ vertices. The two players take turns choosing edges within $K_{n}$, building up a simple graph. The edges must be chosen according to a set of restrictions $\mathcal{R}$. The winner is the last player to choose an edge that does not violate any of the restrictions in $\mathcal{R}$. For fixed $n$ and $\mathcal{R}$, one of the players has a winning strategy. For a pair of games where $\mathcal{R}$ includes bounded degree, connectedness, and triangle-avoidance, we determine the winner for all values of $n$.

The Harmonious Chromatic Number of Bounded Degree Trees

1996 ◽  
Vol 5 (1) ◽  
pp. 15-28 ◽  
Author(s):  
Keith Edwards
Keyword(s):  
Positive Integer ◽  
Chromatic Number ◽  
Maximum Degree ◽  
Polynomial Time Algorithm ◽  
Time Algorithm ◽  
Simple Graph ◽  
Bounded Degree ◽  
Bounded Degree Trees ◽  
Fixed Positive Integer ◽  
Proper Vertex

A harmonious colouring of a simple graph G is a proper vertex colouring such that each pair of colours appears together on at most one edge. The harmonious chromatic number h(G) is the least number of colours in such a colouring. Let d be a fixed positive integer. We show that there is a natural number N(d) such that if T is any tree with m ≥ N(d) edges and maximum degree at most d, then the harmonious chromatic number h(T) is k or k + 1, where k is the least positive integer such that . We also give a polynomial time algorithm for determining the harmonious chromatic number of a tree with maximum degree at most d.

A Winning Strategy for the Ramsey Graph Game

1996 ◽  
Vol 5 (3) ◽  
pp. 267-276 ◽  
Author(s):  
Aleksandar Pekeč
Keyword(s):  
Winning Strategy ◽  
Combinatorial Games ◽  
Graph Game ◽  
Winning Strategies

We consider a «Maker-Breaker’ version of the Ramsey Graph Game, RG(n), and present a winning strategy for Maker requiring at most (n − 3)2n−1 + n + 1 moves. This is the fastest winning strategy known so far. We also demonstrate how the ideas presented can be used to develop winning strategies for some related combinatorial games.

scholarly journals The signless Laplacian spectral radius of bounded degree graphs on surfaces

2015 ◽  
Vol 9 (2) ◽  
pp. 332-346 ◽  
Author(s):  
Guihai Yu ◽  
Lihua Feng ◽  
Aleksandar Ilic ◽  
Dragan Stevanovic
Keyword(s):  
Spectral Radius ◽  
Simple Graph ◽  
Extremal Graphs ◽  
Outerplanar Graphs ◽  
Bounded Degree ◽  
Laplacian Spectral Radius ◽  
Signless Laplacian Spectral Radius ◽  
Signless Laplacian ◽  
Graphs On Surfaces ◽  
Better Than

Let G be an n-vertex (n ? 3) simple graph embeddable on a surface of Euler genus (the number of crosscaps plus twice the number of handles). In this paper, we present upper bounds for the signless Laplacian spectral radius of planar graphs, outerplanar graphs and Halin graphs, respectively, in terms of order and maximum degree. We also demonstrate that our bounds are sometimes better than known ones. For outerplanar graphs without internal triangles, we determine the extremal graphs with the maximum and minimum signless Laplacian spectral radii.

scholarly journals The Topology of Competitively Constructed Graphs

10.37236/3942 ◽  
2014 ◽  
Vol 21 (2) ◽  
Author(s):  
Alan Frieze ◽  
Wesley Pegden
Keyword(s):  
Regular Graph ◽  
Simple Game ◽  
Empty Graph ◽  
Graph Game ◽  
Large Clique

We consider a simple game, the $k$-regular graph game, in which players take turns adding edges to an initially empty graph subject to the constraint that the degrees of vertices cannot exceed $k$. We show a sharp topological threshold for this game: for the case $k=3$ a player can ensure the resulting graph is planar, while for the case $k=4$, a player can force the appearance of arbitrarily large clique minors.

Winning the Argument and Moving the Fight: The Legacy of a Grassroots Humanitarian

Public Voices ◽  
2016 ◽  
Vol 14 (1) ◽  
pp. 115
Author(s):  
Mary Coleman
Keyword(s):  
Higher Education ◽  
United States ◽  
Winning Strategy ◽  
Supreme Court Decision ◽  
The United States ◽  
Colleges And Universities ◽  
Lessons Learned ◽  
The Political ◽  
Political Actors ◽  
Trade Offs

The author of this article argues that the two-decades-long litigation struggle was necessary to push the political actors in Mississippi into a more virtuous than vicious legal/political negotiation. The second and related argument, however, is that neither the 1992 United States Supreme Court decision in Fordice nor the negotiation provided an adequate riposte to plaintiffs’ claims. The author shows that their chief counsel for the first phase of the litigation wanted equality of opportunity for historically black colleges and universities (HBCUs), as did the plaintiffs. In the course of explicating the role of a legal grass-roots humanitarian, Coleman suggests lessons learned and trade-offs from that case/negotiation, describing the tradeoffs as part of the political vestiges of legal racism in black public higher education and the need to move HBCUs to a higher level of opportunity at a critical juncture in the life of tuition-dependent colleges and universities in the United States. Throughout the essay the following questions pose themselves: In thinking about the Road to Fordice and to political settlement, would the Justice Department lawyers and the plaintiffs’ lawyers connect at the point of their shared strength? Would the timing of the settlement benefit the plaintiffs and/or the State? Could plaintiffs’ lawyers hold together for the length of the case and move each piece of the case forward in a winning strategy? Who were plaintiffs’ opponents and what was their strategy? With these questions in mind, the author offers an analysis of how the campaign— political/legal arguments and political/legal remedies to remove the vestiges of de jure segregation in higher education—unfolded in Mississippi, with special emphasis on the initiating lawyer in Ayers v. Waller and Fordice, Isaiah Madison

scholarly journals Power graphs and exchange property for resolving sets

2019 ◽  
Vol 17 (1) ◽  
pp. 1303-1309 ◽  
Author(s):  
Ghulam Abbas ◽  
Usman Ali ◽  
Mobeen Munir ◽  
Syed Ahtsham Ul Haq Bokhary ◽  
Shin Min Kang
Keyword(s):  
Present Article ◽  
Linear Algebra ◽  
Finite Groups ◽  
Robot Navigation ◽  
Simple Graph ◽  
Exchange Property ◽  
Metric Dimension ◽  
Sufficient Condition ◽  
Resolving Set ◽  
Dihedral Groups

Abstract Classical applications of resolving sets and metric dimension can be observed in robot navigation, networking and pharmacy. In the present article, a formula for computing the metric dimension of a simple graph wihtout singleton twins is given. A sufficient condition for the graph to have the exchange property for resolving sets is found. Consequently, every minimal resolving set in the graph forms a basis for a matriod in the context of independence defined by Boutin [Determining sets, resolving set and the exchange property, Graphs Combin., 2009, 25, 789-806]. Also, a new way to define a matroid on finite ground is deduced. It is proved that the matroid is strongly base orderable and hence satisfies the conjecture of White [An unique exchange property for bases, Linear Algebra Appl., 1980, 31, 81-91]. As an application, it is shown that the power graphs of some finite groups can define a matroid. Moreover, we also compute the metric dimension of the power graphs of dihedral groups.

scholarly journals Nanoencapsulated Essential Oils with Enhanced Antifungal Activity for Potential Application on Agri-Food, Material and Environmental Fields

Antibiotics ◽  
2021 ◽  
Vol 10 (1) ◽  
pp. 31
Author(s):  
Magdaléna Kapustová ◽  
Giuseppe Granata ◽  
Edoardo Napoli ◽  
Andrea Puškárová ◽  
Mária Bučková ◽  
...  
Keyword(s):  
Antifungal Activity ◽  
Essential Oil ◽  
Essential Oils ◽  
Winning Strategy ◽  
Colloidal Suspensions ◽  
Fungal Species ◽  
Natural Environments ◽  
Food Material ◽  
Minimum Fungicidal Concentration ◽  
Fungal Strains

Nanotechnology is a new frontier of this century that finds applications in various fields of science with important effects on our life and on the environment. Nanoencapsulation of bioactive compounds is a promising topic of nanotechnology. The excessive use of synthetic compounds with antifungal activity has led to the selection of resistant fungal species. In this context, the use of plant essential oils (EOs) with antifungal activity encapsulated in ecofriendly nanosystems could be a new and winning strategy to overcome the problem. We prepared nanoencapsules containing the essential oils of Origanum vulgare (OV) and Thymus capitatus (TC) by the nanoprecipitation method. The colloidal suspensions were characterized for size, polydispersity index (PDI), zeta potential, efficiency of encapsulation (EE) and loading capacity (LC). Finally, the essential oil nanosuspensions were assayed against a panel of fourteen fungal strains belonging to the Ascomycota and Basidiomycota phyla. Our results show that the nanosystems containing thyme and oregano essential oils were active against various fungal strains from natural environments and materials. In particular, the minimum inhibitory concentration (MIC) and minimum fungicidal concentration (MFC) values were two to four times lower than the pure essential oils. The aqueous, ecofriendly essential oil nanosuspensions with broad-spectrum antifungal activity could be a valid alternative to synthetic products, finding interesting applications in the agri-food and environmental fields.

scholarly journals The Irregularity and Modular Irregularity Strength of Fan Graphs

Symmetry ◽  
2021 ◽  
Vol 13 (4) ◽  
pp. 605
Author(s):  
Martin Bača ◽  
Zuzana Kimáková ◽  
Marcela Lascsáková ◽  
Andrea Semaničová-Feňovčíková
Keyword(s):  
Simple Graph ◽  
Isolated Vertex ◽  
Positive Integers ◽  
Irregularity Strength

For a simple graph G with no isolated edges and at most, one isolated vertex, a labeling φ:E(G)→{1,2,…,k} of positive integers to the edges of G is called irregular if the weights of the vertices, defined as wtφ(v)=∑u∈N(v)φ(uv), are all different. The irregularity strength of a graph G is known as the maximal integer k, minimized over all irregular labelings, and is set to ∞ if no such labeling exists. In this paper, we determine the exact value of the irregularity strength and the modular irregularity strength of fan graphs.

scholarly journals A Lower Bound for the Query Phase of Contraction Hierarchies and Hub Labels and a Provably Optimal Instance-Based Schema

Algorithms ◽  
2021 ◽  
Vol 14 (6) ◽  
pp. 164
Author(s):  
Tobias Rupp ◽  
Stefan Funke
Keyword(s):  
Lower Bound ◽  
Lower Bounds ◽  
Real World ◽  
Road Network ◽  
Upper And Lower Bounds ◽  
Bounded Degree ◽  
Shortest Path Routing ◽  
Graph Classes ◽  
Speed Up ◽  
To Come

We prove a Ω(n) lower bound on the query time for contraction hierarchies (CH) as well as hub labels, two popular speed-up techniques for shortest path routing. Our construction is based on a graph family not too far from subgraphs that occur in real-world road networks, in particular, it is planar and has a bounded degree. Additionally, we borrow ideas from our lower bound proof to come up with instance-based lower bounds for concrete road network instances of moderate size, reaching up to 96% of an upper bound given by a constructed CH. For a variant of our instance-based schema applied to some special graph classes, we can even show matching upper and lower bounds.

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