Discrepancy and large dense monochromatic subsets

A graph is 1-planar if it can be drawn in the plane so that each edge is crossed by at most one other edge. In this paper, it is shown that each 1-planar graph with minimum degree 7 contains a copy of K2∨(K1∪K2) with all vertices of degree at most 12. In addition, we also prove the existence of a graph K1∨(K1∪K2) with relatively small degree vertices in 1-planar graphs with minimum degree at least 6.

In a network cliques are fully connected subgraphs that reveal which are the tight communities present in it. Cliques of size c > 3 are present in random Erdös and Renyi graphs only in the limit of diverging average connectivity.Starting from the finding that real scale-free graphs have large cliques, we study the clique number in uncorrelated scale-free networks finding both upper and lower bounds. Interestingly, we find that in scale-free networks large cliques appear also when the average degree is finite, i.e. even for networks with power law degree distribution exponents γ∊(2,3). Moreover, as long as γ < 3, scale-free networks have a maximal clique which diverges with the system size.

formula omitted]-restricted edge-connectivity in triangle-free graphs

A note on balanced bipartitions

Locally self-avoiding Eulerian tours

Edge vulnerability parameters of split graphs

Super-pancyclic hypergraphs and bipartite graphs

Cops and Robber on some families of oriented graphs

Markovian approach to tackle competing pathogens in simplicial complex

ملخص الدراسة هدفت الدراسة الحالية إلى التعرف على مستوى تحقيق مسار الکليات الإنسانية في السنة التحضيرية بجامعة الملک سعود لأهدافها من وجهة نظر (أعضاء هيئة التدريس، والطلاب)، وقد استخدمت الدراسة المنهج الوصفي التحليلي، والاستبانه لأعضاء هيئة التدريس، والمقابلة مع الطلاب کأداتين للدراسة، وتکونت عينة الدراسة من (65) عضو هيئة تدريس، و(30) طالب.وتوصلت الدراسة إلى أن الأهداف المعرفية والوجدانية بشکل عام تحققت لدى الطلاب بدرجة متوسطة من وجهة نظر أعضاء هيئة التدريس، أما الطلاب فيرون أن الأهداف المعرفية لاتزال ضعيفة لديهم. بينما الاهداف الوجدانية تحققت لديهم بدرجة کبيرة. أما الأهداف النفسحرکية فيرى أعضاء هيئة التدريس أنها تحققت لدى الطلاب بدرجة متوسطة، ولکن المهارات المتعلقة باللغة الانجليزية لم تتحق الا بدرجة ضعيفة، أما الطلاب فيرون أن المهارات المتعلقة بالحاسب الآلي تحققت بدرجة کبيرة بينما المهارات المتعلقة باللغة الانجليزية لاتزال ضعيفة لديهم. Abstract The study aimed at recognizing the level of achieving the Humanitarian Colleges track in the Preparatory Year in King Saud University to their objectives as perceived by faculty members and students. The study used the descriptive method and administered a questionnaire to faculty members and an interview with students. The sample of study consisted of 65 faculty members and 30 students. The results showed that the cognitive and affective objectives were achieved in an average degree as perceived by faculty members, whereas students themselves perceived that the cognitive objectives were low, but they stated that the affective objectives were achieved in high degree. Faculty members stated that psychomotor objectives were achieved in an average degree although some objectives were achieved in a low degree such as English Language skills. Students perceived computer skills as achieved in a high degree, whereas their English Language skills were still low.

We consider a variation of the supermarket model in which the servers can communicate with their neighbors and where the neighborhood relationships are described in terms of a suitable graph. Tasks with unit-exponential service time distributions arrive at each vertex as independent Poisson processes with rate $\lambda$, and each task is irrevocably assigned to the shortest queue among the one it first appears and its $d-1$ randomly selected neighbors. This model has been extensively studied when the underlying graph is a clique in which case it reduces to the well known power-of-$d$ scheme. In particular, results of Mitzenmacher (1996) and Vvedenskaya et al. (1996) show that as the size of the clique gets large, the occupancy process associated with the queue-lengths at the various servers converges to a deterministic limit described by an infinite system of ordinary differential equations (ODE). In this work, we consider settings where the underlying graph need not be a clique and is allowed to be suitably sparse. We show that if the minimum degree approaches infinity (however slowly) as the number of servers $N$ approaches infinity, and the ratio between the maximum degree and the minimum degree in each connected component approaches 1 uniformly, the occupancy process converges to the same system of ODE as the classical supermarket model. In particular, the asymptotic behavior of the occupancy process is insensitive to the precise network topology. We also study the case where the graph sequence is random, with the $N$-th graph given as an Erd\H{o}s-R\'enyi random graph on $N$ vertices with average degree $c(N)$. Annealed convergence of the occupancy process to the same deterministic limit is established under the condition $c(N)\to\infty$, and under a stronger condition $c(N)/\ln N\to\infty$, convergence (in probability) is shown for almost every realization of the random graph.

An immersion of a graph H into a graph G is a one-to-one mapping f: V (H) → V (G) and a collection of edge-disjoint paths in G, one for each edge of H, such that the path P uv corresponding to edge uv has endpoints f(u) and f(v). The immersion is strong if the paths P uv are internally disjoint from f(V (H)). It is proved that for every positive integer Ht, every simple graph of minimum degree at least 200t contains a strong immersion of the complete graph K t . For dense graphs one can say even more. If the graph has order n and has 2cn 2 edges, then there is a strong immersion of the complete graph on at least c 2 n vertices in G in which each path P uv is of length 2. As an application of these results, we resolve a problem raised by Paul Seymour by proving that the line graph of every simple graph with average degree d has a clique minor of order at least cd 3/2, where c>0 is an absolute constant. For small values of t, 1≤t≤7, every simple graph of minimum degree at least t−1 contains an immersion of K t (Lescure and Meyniel [13], DeVos et al. [6]). We provide a general class of examples showing that this does not hold when t is large.

We study the worst-case performance of the maximal matching heuristic applied to the Minimum Vertex Cover and Minimum Maximal Matching problems, through a careful analysis of tight examples. We show that the tight worst-case approximation ratio is asymptotic to \({\rm min}\, \{2, 1/(1-\sqrt{1-\epsilon})\}\) for graphs with an average degree at least εn and to min {2, 1/ε} for graphs with a minimum degree at least εn.KeywordsApproximation RatioMinimum DegreeVertex CoverMaximal MatchDense GraphThese keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

A graph is $k$-linked if for every list of $2k$ vertices $\{s_1,{\ldots}\,s_k, t_1,{\ldots}\,t_k\}$, there exist internally disjoint paths $P_1,{\ldots}\, P_k$ such that each $P_i$ is an $s_i,t_i$-path. We consider degree conditions and connectivity conditions sufficient to force a graph to be $k$-linked.Let $D(n,k)$ be the minimum positive integer $d$ such that every $n$-vertex graph with minimum degree at least $d$ is $k$-linked and let $R(n,k)$ be the minimum positive integer $r$ such that every $n$-vertex graph in which the sum of degrees of each pair of non-adjacent vertices is at least $r$ is $k$-linked. The main result of the paper is finding the exact values of $D(n,k)$ and $R(n,k)$ for every $n$ and $k$.Thomas and Wollan [14] used the bound $D(n,k)\leq (n+3k)/2-2$ to give sufficient conditions for a graph to be $k$-linked in terms of connectivity. Our bound allows us to modify the Thomas–Wollan proof slightly to show that every $2k$-connected graph with average degree at least $12k$ is $k$-linked.

Developing further Stein's recent notion of relative end degrees in infinite graphs, we investigate which degree assumptions can force a locally finite graph to contain a given finite minor, or a finite subgraph of given minimum or average degree. This is part of a wider project which seeks to develop an extremal theory of sparse infinite graphs.