scholarly journals Kempe-Locking Configurations

2018 ◽  
Author(s):  
James A. Tilley
Keyword(s):  
Bipartite Graph ◽  
Systematic Search ◽  
Single Edge ◽  
Heuristic Argument ◽  
Bottom Boundary ◽  
Single Vertex ◽  
Planar Triangulation ◽  
Isomorphism Classes ◽  
Connectivity Property ◽  
Proper Subgraph

Existing proofs of the 4-color theorem succeeded by establishing an unavoidable set of reducible configurations. By this device, their authors showed that a minimum counterexample cannot exist. G.D. Birkhoff proved that a minimum counterexample must satisfy a connectivity property that is referred to in modern parlance as internal 6-connectivity. We show that a minimum counterexample must also satisfy a coloring property, one that we call Kempe-locking. We define the terms Kempe-locking configuration and fundamental Kempe-locking configuration. We provide a heuristic argument that a fundamental Kempe-locking configuration must be of low order and then perform a systematic search through isomorphism classes for such configurations. We describe a methodology for analyzing whether an arbitrary planar triangulation is Kempe-locked; it involves deconstructing the triangulation into a stack of configurations with common endpoints and then creating a bipartite graph of coloring possibilities for each configuration in the stack to assess whether certain 2-color paths can be transmitted from the configuration's top boundary to its bottom boundary. All Kempe-locked triangulations we discovered have two features in common: (1) they are Kempe-locked with respect to only a single edge, say $xy$, and (2) they have a Birkhoff diamond with endpoints $x$ and $y$ as a proper subgraph. On the strength of our various investigations, we are led to a plausible conjecture that the Birkhoff diamond is the only fundamental Kempe-locking configuration. If true, this would establish that the connectivity and coloring properties of a minimum counterexample to the 4-color theorem are incompatible. It would also point to the singular importance of a particularly elegant 4-connected triangulation of order 9 that consists of a triangle enclosing a pentagon enclosing a single vertex.

scholarly journals Kempe-Locking Configurations

Mathematics ◽  
2018 ◽  
Vol 6 (12) ◽  
pp. 309 ◽  
Author(s):  
James Tilley
Keyword(s):  
Systematic Search ◽  
The Novel ◽  
Single Edge ◽  
Heuristic Argument ◽  
Planar Triangulation ◽  
Isomorphism Classes ◽  
Diamond Configuration ◽  
Low Order

The 4-color theorem was proved by showing that a minimum counterexample cannot exist. Birkhoff demonstrated that a minimum counterexample must be internally 6-connected. We show that a minimum counterexample must also satisfy a coloring property that we call Kempe-locking. The novel idea explored in this article is that the connectivity and coloring properties are incompatible. We describe a methodology for analyzing whether an arbitrary planar triangulation is Kempe-locked. We provide a heuristic argument that a fundamental Kempe-locking configuration must be of low order and then perform a systematic search through isomorphism classes for such configurations. All Kempe-locked triangulations that we discovered have two features in common: (1) they are Kempe-locked with respect to only a single edge, say x y , and (2) they have a Birkhoff diamond with endpoints x and y as a subgraph. On the strength of our investigations, we formulate a plausible conjecture that the Birkhoff diamond is the only fundamental Kempe-locking configuration. If true, this would establish that the connectivity and coloring properties of a minimum counterexample are indeed incompatible. It would also imply the appealing conclusion that the Birkhoff diamond configuration alone is responsible for the 4-colorability of planar triangulations.

scholarly journals Spectral Gap of the Largest Eigenvalue of the Normalized Graph Laplacian

2021 ◽  
Author(s):  
Jürgen Jost ◽  
Raffaella Mulas ◽  
Florentin Münch
Keyword(s):  
Lower Bound ◽  
Bipartite Graph ◽  
Spectral Gap ◽  
Complete Bipartite Graph ◽  
Graph Laplacian ◽  
New Method ◽  
Single Edge ◽  
Largest Eigenvalue ◽  
Complement Graph ◽  
The Largest Eigenvalue

AbstractWe offer a new method for proving that the maxima eigenvalue of the normalized graph Laplacian of a graph with n vertices is at least $$\frac{n+1}{n-1}$$ n + 1 n - 1 provided the graph is not complete and that equality is attained if and only if the complement graph is a single edge or a complete bipartite graph with both parts of size $$\frac{n-1}{2}$$ n - 1 2 . With the same method, we also prove a new lower bound to the largest eigenvalue in terms of the minimum vertex degree, provided this is at most $$\frac{n-1}{2}$$ n - 1 2 .

scholarly journals Genus Distributions for Iterated Claws

10.37236/2278 ◽  
2014 ◽  
Vol 21 (1) ◽  
Author(s):  
Jonathan L. Gross ◽  
Imran F. Khan ◽  
Mehvish I. Poshni
Keyword(s):  
Multiple Roots ◽  
Single Edge ◽  
Root Vertex ◽  
Single Vertex ◽  
Single Root

We derive a recursion for the genus distributions of the graphs obtained by iteratively attaching a claw to the dipole $D_3$. The minimum genus of the graphs in this sequence grows arbitrarily large. The families of graphs whose genus distributions have been calculated previously are either planar or almost planar, or they can be obtained by iterative single-vertex or single-edge amalgamation of small graphs. A significant simplifying construction within this calculation achieves the effect of an amalgamation at three vertices with a single root vertex, rather than with multiple roots. 

Note on T-Minimal Complete Bipartite Graphs

1968 ◽  
Vol 11 (5) ◽  
pp. 729-732 ◽  
Author(s):  
I. Z. Bouwer ◽  
I. Broere
Keyword(s):  
Natural Number ◽  
Bipartite Graph ◽  
Complete Bipartite Graph ◽  
Bipartite Graphs ◽  
Different Types ◽  
Complete Bipartite Graphs ◽  
Planar Subgraphs ◽  
Proper Subgraph

The thickness of a graph G is the smallest natural number t such that G is the union of t planar subgraphs. A graph G is t-minimal if its thickness is t and if every proper subgraph of G has thickness < t. (These terms were introduced by Tutte in [3]. In [1, p. 51] Beineke employs the term t-critical instead of t-minimal.) The complete bipartite graph K(m, n) consists of m 'dark1 points, n 'light' points, and the mn lines joining points of different types.

scholarly journals Bipartite graph bundles with connected fibres

1999 ◽  
Vol 59 (1) ◽  
pp. 153-161 ◽  
Author(s):  
Sungpyo Hong ◽  
Jin Ho Kwak ◽  
Jaeun Lee
Keyword(s):  
Bipartite Graph ◽  
Simple Graph ◽  
Isomorphism Classes ◽  
Graph Coverings

Let G be a finite connected simple graph. The isomorphism classes of graph bundles and graph coverings over G have been enumerated by Kwak and Lee. Recently, Archdeacon and others characterised bipartite coverings of G and enumerated the isomorphism classes of regular 2p-fold bipartite coverings of G, when G is nonbipartite. In this paper, we characterise bipartite graph bundles over G and derive some enumeration formulas of the isomorphism classes of them when the fibre is a connected bipartite graph. As an application, we compute the exact numbers of the isomorphism classes of bipartite graph bundles over G when the fibre is the path Pn or the cycle Cn.

scholarly journals Longest paths in random Apollonian networks and largest r-ary subtrees of random d-ary recursive trees

2016 ◽  
Vol 53 (3) ◽  
pp. 846-856 ◽  
Author(s):  
Andrea Collevecchio ◽  
Abbas Mehrabian ◽  
Nick Wormald
Keyword(s):  
Related Result ◽  
Plane Graph ◽  
Time Step ◽  
Single Vertex ◽  
Planar Triangulation ◽  
The Face ◽  
Longest Paths ◽  
Recursive Trees ◽  
Positive Integers ◽  
Iterative Construction

AbstractLet r and d be positive integers with r<d. Consider a random d-ary tree constructed as follows. Start with a single vertex, and in each time-step choose a uniformly random leaf and give it d newly created offspring. Let 𝒯d,t be the tree produced after t steps. We show that there exists a fixed δ<1 depending on d and r such that almost surely for all large t, every r-ary subtree of 𝒯d,t has less than tδ vertices. The proof involves analysis that also yields a related result. Consider the following iterative construction of a random planar triangulation. Start with a triangle embedded in the plane. In each step, choose a bounded face uniformly at random, add a vertex inside that face and join it to the vertices of the face. In this way, one face is destroyed and three new faces are created. After t steps, we obtain a random triangulated plane graph with t+3 vertices, which is called a random Apollonian network. We prove that there exists a fixed δ<1, such that eventually every path in this graph has length less than t𝛿, which verifies a conjecture of Cooper and Frieze (2015).

scholarly journals On Factors of Independent Transversals in $k$-Partite Graphs

2021 ◽  
Vol 28 (4) ◽  
Author(s):  
Raphael Yuster
Keyword(s):  
Bipartite Graph ◽  
Greedy Algorithm ◽  
Pairwise Disjoint ◽  
Independent Set ◽  
Single Vertex ◽  
Hall's Theorem ◽  
Hall’S Theorem ◽  
The Greedy Algorithm

A $[k,n,1]$-graph is a $k$-partite graph with parts of order $n$ such that the bipartite graph induced by any pair of parts is a matching. An independent transversal in such a graph is an independent set that intersects each part in a single vertex. A factor of independent transversals is a set of $n$ pairwise-disjoint independent transversals. Let $f(k)$ be the smallest integer $n_0$ such that every $[k,n,1]$-graph has a factor of independent transversals assuming $n \geqslant n_0$. Several known conjectures imply that for $k \geqslant 2$, $f(k)=k$ if $k$ is even and $f(k)=k+1$ if $k$ is odd. While a simple greedy algorithm based on iterating Hall's Theorem shows that $f(k) \leqslant 2k-2$, no better bound is known and in fact, there are instances showing that the bound $2k-2$ is tight for the greedy algorithm. Here we significantly improve upon the greedy algorithm bound and prove that $f(k) \leqslant 1.78k$ for all $k$ sufficiently large, answering a question of MacKeigan.

scholarly journals Hogyan befolyásolja a villamosenergia-hálózatról rendelkezésre álló információ a fizikai támadások által okozott sérülékenységről alkotott képet?

2021 ◽  
Vol 2 (2) ◽  
pp. 155-163
Author(s):  
Bálint Hartmann
Keyword(s):  
Power Grid ◽  
Deep Understanding ◽  
Graph Representation ◽  
Single Edge ◽  
Weighted Network ◽  
Single Vertex ◽  
Targeted Attacks ◽  
Edge Removal ◽  
Element Removal ◽  
Vertex Removal

Összefoglaló. A villamosenergia-rendszerek fizikai támadásokkal szembeni ellenálló képessége a közelmúltban világszerte történt események ismeretében egyre nagyobb hangsúlyt kap a tématerület kutatásaiban. Az ilyen eseményekre való megfelelő felkészüléshez elengedhetetlen az üzemeltetett infrastruktúrának, elsősorban annak gyengeségeinek pontos ismerete. A cikkben Magyarország villamosenergia-hálózatának adatai alapján készített súlyozatlan és súlyozott gráfokon végzünk vizsgálatokat, hogy megértsük a különböző stratégia mentén kiválasztott célpontok elleni támadások milyen mértékben csökkentik a topológiai hatékonyságot. A cikk célja egyben a magyar hálózat sérülékenységének általános bemutatása is, mely hasznos bemeneti információ lehet a kockázati tervek elkészítésekor. Summary. Tolerance of the power grid against physical intrusions has gained importance in the light of various attacks that have taken place around the world. To adequately prepare for such events, grid operators have to possess a deep understanding of their infrastructure, more specifically, of its weaknesses. A graph representation of the Hungarian power grid was created in a way that the vertices are generators, transformers, and substations and the edges are high-voltage transmission lines. All transmission and sub-transmission elements were considered, including the 132 kV network as well. The network is subjected to various types of single and double element attacks, objects of which are selected according to different aspects. The vulnerability of the network is measured as a relative drop in efficiency when a vertex or an edge is removed from the network. Efficiency is a measure of the network’s performance, assuming that the efficiency for transmitting electricity between vertices i and j is proportional to the reciprocal of their distance. In this paper, simultaneous removals were considered, arranged into two scenarios (single or double element removal) and a total of 5 cases were carried out (single vertex removal, single edge removal, double vertex removal, double edge removal, single vertex and single edge removal). During the examinations, all possible removal combinations were simulated, thus the 5 cases represent 385, 504, 73920, 128271 and 193797 runs, respectively. After all runs were performed, damage values were determined for random and targeted attacks, and attacks causing maximal damage were also identified. In all cases, damage was calculated for unweighted and weighted networks as well, to enable the comparison of those two models. The aims of this paper are threefold: to perform a general assessment on the vulnerability of the Hungarian power grid against random and targeted attacks; to compare the damage caused by different attack strategies; and to highlight the differences between using unweighted and weighted graphs representations. Random removal of a single vertex or a single edge caused 0.3–0.4% drop in efficiency, respectively, which indicates a high tolerance against such attacks. Damage for random double attacks was still only in the range of 0.6–0.8%, which is acceptable. It was shown that if targets are selected by the attacker based on the betweenness rank of the element, damage would be below the maximal possible values. Comparison of the damage measured in the unweighted and the weighted network representations has shown that damage to the weighted network tends to be bigger for vertex attacks, but the contrary is observed for edge attacks. Numerical differences between the two representations do not show any trend that could be generalised, but in the case of the most vulnerable elements significant differences were found in damage measures, which underlines the importance of using weighted models.

scholarly journals Customer Loyalty Improves the Effectiveness of Recommender Systems Based on Complex Network

2020 ◽  
Author(s):  
Yun Bai ◽  
Suling Jia ◽  
Shuangzhe Wang ◽  
Binkai Tan
Keyword(s):  
Bipartite Graph ◽  
Customer Loyalty ◽  
Recommender System ◽  
Dimensional Space ◽  
Recommendation Algorithm ◽  
Single Vertex ◽  
Recommendation Algorithms ◽  
Vertex Set ◽  
Time And Energy ◽  
Better Than

A good recommender system can infer customers&rsquo; preferences based on their historical purchase records, and recommend products that the customers may be interested in, saving them a lot of time and energy. For enterprises, it is difficult to recommend accurately to each customer, and the bad recommendation may be counterproductive. Customer loyalty is an indicator that measures the preference relationship between customers and products in the field of marketing. A hypothesis is proposed in this study: if companies can divide customers into different groups based on customer loyalty, the recommendation effect on certain groups is better than that on overall customers. In this study, customer loyalty is measured by four features of the RFML model. All customers are viewed as points on a four-dimensional space, which are clustered by the k-means model. Two recommendation algorithms based on complex networks are tested: recommendation algorithm based on bipartite graph and PersonalRank (BGPR), and recommendation algorithm based on a single vertex set network and DeepWalk (SVDW). The experimental results show that customer loyalty has improved the effectiveness of the two algorithms over 14%, and the recommendation effect is the best on customer groups with a loyalty level of 4 (the highest level is 5). The recommendation algorithms with customer loyalty are better than using them alone.

scholarly journals Minimum Weight $H$-Decompositions of Graphs: The Bipartite Case

10.37236/613 ◽  
2011 ◽  
Vol 18 (1) ◽  
Author(s):  
Teresa Sousa
Keyword(s):  
Bipartite Graph ◽  
Minimum Weight ◽  
Total Weight ◽  
Combinatorial Theory ◽  
Single Edge ◽  
Asymptotic Value ◽  
Large N ◽  
Bipartite Case

Given graphs $G$ and $H$ and a positive number $b$, a weighted $(H,b)$-decomposition of $G$ is a partition of the edge set of $G$ such that each part is either a single edge or forms an $H$-subgraph. We assign a weight of $b$ to each $H$-subgraph in the decomposition and a weight of 1 to single edges. The total weight of the decomposition is the sum of the weights of all elements in the decomposition. Let $\phi(n,H,b)$ be the the smallest number such that any graph $G$ of order $n$ admits an $(H,b)$-decomposition with weight at most $\phi(n,H,b)$. The value of the function $\phi(n,H,b)$ when $b=1$ was determined, for large $n$, by Pikhurko and Sousa [Minimum $H$-Decompositions of Graphs, Journal of Combinatorial Theory, B, 97 (2007), 1041–1055.] Here we determine the asymptotic value of $\phi(n,H,b)$ for any fixed bipartite graph $H$ and any value of $b$ as $n$ tends to infinity.

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