A note on dominating cycles in 2-connected graphs

Let G be a 2-connected graph on n vertices such that d(x) + d(y) + d(z) n for all triples of independent vertices x, y, z. We prove that every longest cycle in G is a dominating cycle unless G is a spanning subgraph of a graph belonging to one of four easily specified classes of graphs

Qualitative Research on Youths’ Social Media Use: A review of the literature

In this article we explore how educational researchers report empirical qualitative research about young people’s social media use. We frame the overall study with an understanding that social media sites contribute to the production of neoliberal subjects, and we draw on Foucauldian discourse theories and the understanding that how researchers explain topics and concepts produces particular ways of thinking about the world while excluding others. Findings include that 1) there is an absence of attention to the structure and function of social media platforms; 2) adolescents are positioned in problematic, developmental ways, and 3) the over-representation of girls and young women in these studies contributes to the feminization of problems on social media. We conclude by calling for future research that can serve as a robust resource for exploring adolescents’ social media use in more productive, nuanced ways

Polynomial algorithms that prove an NP-hard hypothesis implies an NP-hard conclusion

A number of results in Hamiltonian graph theory are of the form $\mathcal{P}$$_{1}$ implies $\mathcal{P}$$_{2}$, where $\mathcal{P}$$_{1}$ is a property of graphs that is NP-hard and $\mathcal{P}$$_{2}$ is a cycle structure property of graphs that is also NP-hard. Such a theorem is the well-known Chv\'{a}tal-Erd\"{o}s Theorem, which states that every graph $G$ with $\alpha \leq \kappa$ is Hamiltonian. Here $\kappa$ is the vertex connectivity of $G$ and $\alpha$ is the cardinality of a largest set of independent vertices of $G$. In another paper Chv\'{a}tal points out that the proof of this result is in fact a polynomial time construction that either produces a Hamilton cycle or a set of more than $\kappa$ independent vertices. In this note we point out that other theorems in Hamiltonian graph theory have a similar character. In particular, we present a constructive proof of the well-known theorem of Jung for graphs on $16$ or more vertices.. \u

Degree Sequences and the Existence of kk-Factors

We consider sufficient conditions for a degree sequence $\pi$ to be forcibly $k$-factor graphical. We note that previous work on degrees and factors has focused primarily on finding conditions for a degree sequence to be potentially $k$-factor graphical. We first give a theorem for $\pi$ to be forcibly 1-factor graphical and, more generally, forcibly graphical with deficiency at most $\beta\ge0$. These theorems are equal in strength to Chv\'atal's well-known hamiltonian theorem, i.e., the best monotone degree condition for hamiltonicity. We then give an equally strong theorem for $\pi$ to be forcibly 2-factor graphical. Unfortunately, the number of nonredundant conditions that must be checked increases significantly in moving from $k=1$ to $k=2$, and we conjecture that the number of nonredundant conditions in a best monotone theorem for a $k$-factor will increase superpolynomially in $k$. This suggests the desirability of finding a theorem for $\pi$ to be forcibly $k$-factor graphical whose algorithmic complexity grows more slowly. In the final section, we present such a theorem for any $k\ge2$, based on Tutte's well-known factor theorem. While this theorem is not best monotone, we show that it is nevertheless tight in a precise way, and give examples illustrating this tightness.Comment: 19 page

Orienting Graphs to Optimize Reachability

The paper focuses on two problems: (i) how to orient the edges of an undirected graph in order to maximize the number of ordered vertex pairs (x,y) such that there is a directed path from x to y, and (ii) how to orient the edges so as to minimize the number of such pairs. The paper describes a quadratic-time algorithm for the first problem, and a proof that the second problem is NP-hard to approximate within some constant 1+epsilon > 1. The latter proof also shows that the second problem is equivalent to ``comparability graph completion''; neither problem was previously known to be NP-hard

Best monotone degree conditions for binding number

AbstractWe give sufficient conditions on the vertex degrees of a graph G to guarantee that G has binding number at least b, for any given b>0. Our conditions are best possible in exactly the same way that Chvátal’s well-known degree condition to guarantee a graph is Hamiltonian is best possible

A Survey of Best Monotone Degree Conditions for Graph Properties

We survey sufficient degree conditions, for a variety of graph properties, that are best possible in the same sense that Chvatal's well-known degree condition for hamiltonicity is best possible.Comment: 25 page

Инженерно-геологические условия Майминского района и проект инженерно-геологических изысканий под реконструкцию аэровокзального комплекса аэропорта (Республика Алтай)

Настоящая работа представляет собой проект инженерно-геологических изысканий под реконструкцию аэровокзального комплекса аэропорта Горно-Алтайск. Целью проектирования является изучение инженерно-геологических условий участка и разработка проекта инженерно-геологических изысканий под реконструкцию аэровокзального комплекса аэропорта Горно-Алтайск.This work is a project of engineering and geological surveys for the reconstruction of the airport terminal complex Gorno-Altaysk. The purpose of the design is to study the engineering and geological conditions of the site and develop a project of engineering and geological surveys for the reconstruction of the airport terminal complex Gorno-Altaysk