Vertices Of Small Degree Research Articles

On conflict-free proper colourings of graphs without small degree vertices

A proper vertex colouring of a graph G is conflict-free if in the neighbourhood of every vertex some colour appears exactly once, while it is called h-conflict-free if there are at least h such colours for each vertex of G. The least numbers of colours in such colourings of G are denoted by χpcf(G) and χpcfh(G), respectively. The latter parameter may be regarded as a natural relaxation of the 2-chromatic number, χ2(G), i.e. the least number of colours in a proper colouring of the square of a given graph G. It is known that χpcfh(G) can be as large as (h+1)(Δ+1)≈Δ2 for graphs with maximum degree Δ and h very close to Δ. We provide several new upper bounds for these parameters for graphs with minimum degrees δ large enough and h of smaller order than δ. In particular, we show that χpcfh(G)⩽(1+o(1))Δ if δ≫ln⁡Δ and h≪δ, and that χpcf(G)⩽Δ+O(ln⁡Δ) for regular graphs. These are related with the conjecture of Caro, Petruševski and Škrekovski that χpcf(G)⩽Δ+1 for every connected graph G of maximum degree Δ⩾3, towards which they proved that χpcf(G)⩽⌊5Δ2⌋ if Δ⩾1.

Minimal Ramsey Graphs with Many Vertices of Small Degree

Given any graph $H$, a graph $G$ is said to be $q$-Ramsey for $H$ if every coloring of the edges of $G$ with $q$ colors yields a monochromatic subgraph isomorphic to $H$. Such a graph $G$ is said to be minimal $q$-Ramsey for $H$ if additionally no proper subgraph $G'$ of $G$ is $q$-Ramsey for $H$. In 1976, Burr, Erdös, and Lovász initiated the study of the parameter $s_q(H)$, defined as the smallest minimum degree among all minimal $q$-Ramsey graphs for $H$. In this paper, we consider the problem of determining how many vertices of degree $s_q(H)$ a minimal $q$-Ramsey graph for $H$ can contain. Specifically, we seek to identify graphs for which a minimal $q$-Ramsey graph can contain arbitrarily many such vertices. We call a graph satisfying this property $s_q$-abundant. Among other results, we prove that every cycle is $s_q$-abundant for any integer $q\geq 2$. We also discuss the cases when $H$ is a clique or a clique with a pendant edge, extending previous results of Burr and co-authors and Fox and co-authors. To prove our results and construct suitable minimal Ramsey graphs, we use gadget graphs, which we call pattern gadgets and which generalize earlier constructions used in the study of minimal Ramsey graphs. We provide a new, more constructive proof of the existence of these gadgets.

On measuring network robustness for weighted networks

Network robustness measures how well network structure is strong and healthy when it is under attack, such as vertices joining and leaving. It has been widely used in many applications, such as information diffusion, disease transmission, and network security. However, existing metrics, including node connectivity, edge connectivity, and graph expansion, can be suboptimal for measuring network robustness since they are inefficient to be computed and cannot directly apply to the weighted networks or disconnected networks. In this paper, we define the \({\mathcal {R}}\)-energy as a new robustness measurement for weighted networks based on the method of spectral analysis. \({\mathcal {R}}\)-energy can cope with disconnected networks and is efficient to compute with a time complexity of \(O(|V|+|E|)\), where V and E are sets of vertices and edges in the network, respectively. Our experiments illustrate the rationality and efficiency of computing \({\mathcal {R}}\)-energy: (1) Removal of high degree vertices reduces network robustness more than that of random or small degree vertices; (2) it takes as little as 120 s to compute for a network with about 6M vertices and 33M edges. We can further detect events occurring in a dynamic Twitter network with about 130K users and discover interesting weekly tweeting trends by tracking changes to \({\mathcal {R}}\)-energy.

Efficient Method for Improving the Spreading Efficiency in Small-World Networks and Assortative Scale-Free Networks

Many complex systems are abstractly regarded as complex networks in the study of complicated problems. The research field of information spreading in complex networks has attracted extensive interest. Many spreading strategies have been proposed for improving the spreading efficiency in complex networks. However, the strategies differ in terms of performance in various complex networks. In this paper, a hybrid and effective method for improving the spreading efficiency in small-world networks and assortative scale-free networks is proposed. The proposed method can be applied to solve the essential problem of low spreading efficiency due to spreading to small-degree vertices. The proposed method combines two strategies: 1) a set of top small-degree vertices are specified as the initial spreaders and 2) vertices preferentially spread information to large-degree neighbors. Sixty-eight groups of Monte Carlo experiments are conducted in three real complex networks and seventeen synthetic complex networks. According to the experimental results and theoretical analysis, the proposed method is efficient for improving the spreading efficiency in small-world networks and assortative scale-free networks. Moreover, in assortative scale-free networks, the improvement in the spreading efficiency that is realized via the proposed method increases with the assortativity coefficient.

Designing efficient hybrid strategies for information spreading in scale-free networks.

Designing a spreading strategy is one of the critical issues strongly affecting spreading efficiency in complex networks. In this paper, to improve the efficiency of information spreading in scale-free networks, we propose four hybrid strategies by combining two basic strategies, i.e. (i) the LS (in which information is preferentially spread from the large-degree vertices to the small-degree ones), and (ii) the SL (in which information is preferentially spread from the small-degree vertices to the large-degree ones). The objective in combining the two basic LS and SL strategies is to fully exploit the advantages of both strategies. To evaluate the spreading efficiency of the proposed four hybrid strategies, we first propose an information spreading model. Then, we introduce the details of the proposed hybrid strategies that are formulated by combining LS and SL. Third, we build a set of scale-free network structures by differently configuring the relevant parameters. In addition, finally, we conduct various Monte Carlo experiments to examine the spreading efficiency of the proposed hybrid strategies in different scale-free network structures. Experimental results indicate that the proposed hybrid strategies are effective and efficient for spreading information in scale-free networks.

On the approximation of Laplacian eigenvalues in graph disaggregation

Graph disaggregation is a technique used to address the high cost of computation for power law graphs on parallel processors. The few high-degree vertices are broken into multiple small-degree vertices, in order to allow for more efficient computation in parallel. In particular, we consider computations involving the graph Laplacian, which has significant applications, including diffusion mapping and graph partitioning, among others. We prove results regarding the spectral approximation of the Laplacian of the original graph by the Laplacian of the disaggregated graph. In addition, we construct an alternate disaggregation operator whose eigenvalues interlace those of the original Laplacian. Using this alternate operator, we construct a uniform preconditioner for the original graph Laplacian.

A note on the annihilation number and 2-domination number of a tree

In 2014, Desormeaux et al. (Discrete Math 319:15---23, 2014) proved a relationship between the annihilation number and 2-domination number of a tree. In this note, we provide a family of bounds for the 2-domination number of a tree based on the amount of vertices of small degree. This family of bounds extends current bounds on the 2-domination number of a tree, and provides an alternative proof for the relationship between the annihilation number and the 2-domination number of a tree that was shown by Desormeaux et al.

Detecting community structures in networks by label propagation with prediction of percolation transition.

Though label propagation algorithm (LPA) is one of the fastest algorithms for community detection in complex networks, the problem of trivial solutions frequently occurring in the algorithm affects its performance. We propose a label propagation algorithm with prediction of percolation transition (LPAp). After analyzing the reason for multiple solutions of LPA, by transforming the process of community detection into network construction process, a trivial solution in label propagation is considered as a giant component in the percolation transition. We add a prediction process of percolation transition in label propagation to delay the occurrence of trivial solutions, which makes small communities easier to be found. We also give an incomplete update condition which considers both neighbor purity and the contribution of small degree vertices to community detection to reduce the computation time of LPAp. Numerical tests are conducted. Experimental results on synthetic networks and real-world networks show that the LPAp is more accurate, more sensitive to small community, and has the ability to identify a single community structure. Moreover, LPAp with the incomplete update process can use less computation time than LPA, nearly without modularity loss.

Small degree vertices in minimal bricks

Bricks are 3-connected bicritical graphs. Together with braces, they arise in matching theory as building blocks for matching covered graphs.We prove that every minimal brick on n vertices has at least n9 vertices of degree at most 4. This proves a relaxed version of a conjecture of Norine and Thomas, which suggests that every minimal brick has some fixed positive fraction of vertices of degree 3.

Minimal bricks have many vertices of small degree

We prove that every minimal brick on n vertices has at least n/9 vertices of degree at most 4.

Light subgraphs in the family of 1-planar graphs with high minimum degree

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.

On local properties of 1-planar graphs with high minimum degree

A graph is called 1-planar if there exists its drawing in the plane such that each edge contains at most one crossing. We prove that each 1-planar graph of minimum degree 7 contains a pair of adjacent vertices of degree 7 as well as several small graphs whose vertices have small degrees; we also prove the existence of a 4-cycle with relatively small degree vertices in 1-planar graphs of minimum degree at least 6.

Lower Bound of the Number of Maximum Genus Embeddings and Genus Embeddings of K 12s+7

In this paper, we study lower bound of the number of maximum orientable genus embeddings (or MGE in short) for a loopless graph. We show that a connected loopless graph of order n has at least \({\frac{1}{4^{\gamma_M(G)}}\prod_{v\in{V(G)}}{(d(v)-1)!}}\) distinct MGE’s, where γM(G) is the maximum orientable genus of G. Infinitely many examples show that this bound is sharp (i.e., best possible) for some types of graphs. Compared with a lower bound of Stahl (Eur J Combin 13:119–126, 1991) which concerns upper-embeddable graphs (i.e., embedded graphs with at most two facial walks), this result is more general and effective in the case of (sparse) graphs permitting relative small-degree vertices. We also obtain a similar formula for maximum nonorientable genus embeddings for general graphs. If we apply our orientable results to the current graph Gs of K12s+7, then Gs has at least 23s distinct MGE’s.This implies that K12s+7 has at least (22)s nonisomorphic cyclic oriented triangular embeddings for sufficient large s.

Ends and Vertices of Small Degree in Infinite Minimally k-(Edge)-Connected Graphs

Bounds on the minimum degree and on the number of vertices attaining it have been much studied for finite edge-/vertex-minimally k-connected/k-edge-connected graphs. We give an overview of the results known for finite graphs and show that most of these carry over to infinite graphs if we consider ends of small degree as well as vertices.

Vertices of Small Degree in Uniquely Hamiltonian Graphs

LetGbe a uniquely hamiltonian graph onnvertices. We show thatGhas a vertex of degree at mostc log28n+3, wherec=(2−log2 3)−1≈2.41. We show further thatGhas at least two vertices of degree less than four if it is planar and at least four vertices of degree two if it is bipartite.

Planar graphs with few vertices of small degree

Given n > k > 0, Erdös and Griggs introduced ak(n) = minG | deg(v) < k|, where G runs over all simple planar graphs on n vertices. With many constructions and the Euler formula, we determine ak(n) for all n ⩾ nk: (i) ak(n) = 0, if k < 6; (ii) a6(n) = 4, if n is even; a6(n) = 5, if n is odd; (iii) ak(n) = ⌈ ((k − 6)n + 12)/(k − 3) ⌉, if k = 7,8,9,10; (iv) a11(n) = ⌈ (5n + 18)/8 ⌉; (v)ak(n) = ⌈ ((k − 8)n + 16)/(k − 6)⌉, if k ⩾ 12. The cases k ⩽ 6 follow from results of Grünbaum and Motzkin, while the cases k ⩾ 12 and the lower bound for a11(n) are due to West and Will.

Asymptotic number of symmetries in locally restricted trees

Exact and asymptotic formulas were used to compute the number of symmetries in several types of unlabeled trees with vertices of small degree.

Disjunkte Fragmente in kritisch n-fach zusammenhängenden Graphen

It is proved that every finite, non-complete, critically n -connected graph contains two disjoint fragments F such that | F |≤ n /2. Furthermore, it contains at least four disjoint fragments of any order. These results are best possible and imply almost all known facts on vertices of small degree in critically n -connected graphs.