scholarly journals Distinct Distances in Graph Drawings

10.37236/831 ◽  
2008 ◽  
Vol 15 (1) ◽  
Author(s):  
Paz Carmi ◽  
Vida Dujmović ◽  
Pat Morin ◽  
David R. Wood
Keyword(s):  
Maximum Degree ◽  
Combinatorial Geometry ◽  
Complete Graphs ◽  
Bounded Degree ◽  
Cartesian Products ◽  
Bounded Treewidth ◽  
Line Drawings ◽  
Straight Line ◽  
Minimum Number ◽  
Complete Bipartite Graphs

The distance-number of a graph $G$ is the minimum number of distinct edge-lengths over all straight-line drawings of $G$ in the plane. This definition generalises many well-known concepts in combinatorial geometry. We consider the distance-number of trees, graphs with no $K^-_4$-minor, complete bipartite graphs, complete graphs, and cartesian products. Our main results concern the distance-number of graphs with bounded degree. We prove that $n$-vertex graphs with bounded maximum degree and bounded treewidth have distance-number in ${\cal O}(\log n)$. To conclude such a logarithmic upper bound, both the degree and the treewidth need to be bounded. In particular, we construct graphs with treewidth $2$ and polynomial distance-number. Similarly, we prove that there exist graphs with maximum degree $5$ and arbitrarily large distance-number. Moreover, as $\Delta$ increases the existential lower bound on the distance-number of $\Delta$-regular graphs tends to $\Omega(n^{0.864138})$.

scholarly journals Size Ramsey Number of Bounded Degree Graphs for Games

2013 ◽  
Vol 22 (4) ◽  
pp. 499-516
Author(s):  
HEIDI GEBAUER
Keyword(s):  
Connected Graph ◽  
Maximum Degree ◽  
Ramsey Number ◽  
Bounded Degree ◽  
Minimum Number ◽  
Game Theoretic ◽  
Bounded Degree Graphs ◽  
Constant Maximum

We study Maker/Breaker games on the edges ofsparsegraphs. Maker and Breaker take turns at claiming previously unclaimed edges of a given graphH. Maker aims to occupy a given target graphGand Breaker tries to prevent Maker from achieving his goal. We show that for everydthere is a constantc=c(d)with the property that for every graphGonnvertices of maximum degreedthere is a graphHon at mostcnedges such that Maker has a strategy to occupy a copy ofGin the game onH.This is a result about a game-theoretic variant of the size Ramsey number. For a given graphG,$\hat{r}'(G)$is defined as the smallest numberMfor which there exists a graphHwithMedges such that Maker has a strategy to occupy a copy ofGin the game onH. In this language, our result yields that for every connected graphGof constant maximum degree,$\hat{r}'(G) = \Theta(n)$.Moreover, we can also use our method to settle the corresponding extremal number foruniversalgraphs: for a constantdand for the class${\cal G}_{n}$ofn-vertex graphs of maximum degreed,$s({\cal G}_{n})$denotes the minimum number such that there exists a graphHwithMedges where, foreveryG∈${\cal G}_{n}$, Maker has a strategy to build a copy ofGin the game onH. We obtain that$s({\cal G}_{n}) = \Theta(n^{2 - \frac{2}{d}})$.

scholarly journals Цветовая энергия некоторых кластерных графов

2021 ◽  
pp. 51-64
Author(s):  
S. D'Souza ◽  
K.P. Girija ◽  
H.J. Gowtham
Keyword(s):  
Quantum Chemistry ◽  
Adjacency Matrix ◽  
Connected Graph ◽  
Complete Graphs ◽  
Vertex Deletion ◽  
Minimum Number ◽  
Complete Bipartite Graphs ◽  
Proper Generalization ◽  
Simple Connected Graph ◽  
Cluster Graphs

Let $G$ be a simple connected graph. The energy of a graph $G$ is defined as sum of the absolute eigenvalues of an adjacency matrix of the graph $G$. It represents a proper generalization of a formula valid for the total $\pi$-electron energy of a conjugated hydrocarbon as calculated by the Huckel molecular orbital (HMO) method in quantum chemistry. A coloring of a graph $G$ is a coloring of its vertices such that no two adjacent vertices share the same color. The minimum number of colors needed for the coloring of a graph $G$ is called the chromatic number of $G$ and is denoted by $\chi(G)$. The color energy of a graph $G$ is defined as the sum of absolute values of the color eigenvalues of $G$. The graphs with large number of edges are referred as cluster graphs. Cluster graphs are graphs obtained from complete graphs by deleting few edges according to some criteria. It can be obtained on deleting some edges incident on a vertex, deletion of independent edges/triangles/cliques/path P3 etc. Bipartite cluster graphs are obtained by deleting few edges from complete bipartite graphs according to some rule. In this paper, the color energy of cluster graphs and bipartite cluster graphs are studied.

scholarly journals Bounded-Degree Graphs can have Arbitrarily Large Slope Numbers

10.37236/1139 ◽  
2006 ◽  
Vol 13 (1) ◽  
Author(s):  
János Pach ◽  
Dömötör Pálvölgyi
Keyword(s):  
Maximum Degree ◽  
Line Drawing ◽  
Bounded Degree ◽  
Straight Line ◽  
Bounded Degree Graphs

We construct graphs with $n$ vertices of maximum degree $5$ whose every straight-line drawing in the plane uses edges of at least $n^{1/6-o(1)}$ distinct slopes.

scholarly journals QUANTUM PERFECT STATE TRANSFER ON WEIGHTED JOIN GRAPHS

2009 ◽  
Vol 07 (08) ◽  
pp. 1429-1445 ◽  
Author(s):  
RICARDO JAVIER ANGELES-CANUL ◽  
RACHAEL M. NORTON ◽  
MICHAEL C. OPPERMAN ◽  
CHRISTOPHER C. PARIBELLO ◽  
MATTHEW C. RUSSELL ◽  
...  
Keyword(s):  
Regular Graph ◽  
Complete Graphs ◽  
Perfect State ◽  
Weighted Graphs ◽  
Cartesian Products ◽  
Hamming Graph ◽  
State Transfer ◽  
Vertex Graph ◽  
Complete Bipartite Graphs ◽  
Perfect State Transfer

This paper studies quantum perfect state transfer on weighted graphs. We prove that the join of a weighted two-vertex graph with any regular graph has perfect state transfer. This generalizes a result of Casaccino et al.1 where the regular graph is a complete graph with or without a missing edge. In contrast, we prove that the half-join of a weighted two-vertex graph with any weighted regular graph has no perfect state transfer. As a corollary, unlike for complete graphs, adding weights in complete bipartite graphs does not produce perfect state transfer. We also observe that any Hamming graph has perfect state transfer between each pair of its vertices. The result is a corollary of a closure property on weighted Cartesian products of perfect state transfer graphs. Moreover, on a hypercube, we show that perfect state transfer occurs between uniform superpositions on pairs of arbitrary subcubes, thus generalizing results of Bernasconi et al.2 and Moore and Russell.3

scholarly journals Analogies between the Crossing Number and the Tangle Crossing Number

10.37236/7581 ◽  
2018 ◽  
Vol 25 (4) ◽  
Author(s):  
Robin Anderson ◽  
Shuliang Bai ◽  
Fidel Barrera-Cruz ◽  
Éva Czabarka ◽  
Giordano Da Lozzo ◽  
...  
Keyword(s):  
Perfect Matching ◽  
Crossing Number ◽  
Binary Trees ◽  
Biologically Relevant ◽  
Line Drawings ◽  
Crossing Numbers ◽  
Parallel Lines ◽  
Straight Line ◽  
Minimum Number ◽  
Number Of Leaves

Tanglegrams are special graphs that consist of a pair of rooted binary trees with the same number of leaves, and a perfect matching between the two leaf-sets. These objects are of use in phylogenetics and are represented with straight-line drawings where the leaves of the two plane binary trees are on two parallel lines and only the matching edges can cross. The tangle crossing number of a tanglegram is the minimum number of crossings over all such drawings and is related to biologically relevant quantities, such as the number of times a parasite switched hosts.Our main results for tanglegrams which parallel known theorems for crossing numbers are as follows. The removal of a single matching edge in a tanglegram with $n$ leaves decreases the tangle crossing number by at most $n-3$, and this is sharp. Additionally, if $\gamma(n)$ is the maximum tangle crossing number of a tanglegram with $n$ leaves, we prove $\frac{1}{2}\binom{n}{2}(1-o(1))\le\gamma(n)<\frac{1}{2}\binom{n}{2}$. For an arbitrary tanglegram $T$, the tangle crossing number, $\mathrm{crt}(T)$, is NP-hard to compute (Fernau et al. 2005). We provide an algorithm which lower bounds $\mathrm{crt}(T)$ and runs in $O(n^4)$ time. To demonstrate the strength of the algorithm, simulations on tanglegrams chosen uniformly at random suggest that the tangle crossing number is at least $0.055n^2$ with high probabilty, which matches the result that the tangle crossing number is $\Theta(n^2)$ with high probability (Czabarka et al. 2017).

Acute Constraints in Straight-Line Drawings of Planar Graphs

2019 ◽  
Vol E102.A (9) ◽  
pp. 994-1001
Author(s):  
Akane SETO ◽  
Aleksandar SHURBEVSKI ◽  
Hiroshi NAGAMOCHI ◽  
Peter EADES
Keyword(s):  
Planar Graphs ◽  
Line Drawings ◽  
Straight Line

scholarly journals Clustered 3-colouring graphs of bounded degree

2021 ◽  
pp. 1-13
Author(s):  
Vida Dujmović ◽  
Louis Esperet ◽  
Pat Morin ◽  
Bartosz Walczak ◽  
David R. Wood
Keyword(s):  
Planar Graphs ◽  
Maximum Degree ◽  
Bounded Degree ◽  
Bounded Number ◽  
Proper Vertex ◽  
Vertex Colouring

Abstract A (not necessarily proper) vertex colouring of a graph has clustering c if every monochromatic component has at most c vertices. We prove that planar graphs with maximum degree $\Delta$ are 3-colourable with clustering $O(\Delta^2)$ . The previous best bound was $O(\Delta^{37})$ . This result for planar graphs generalises to graphs that can be drawn on a surface of bounded Euler genus with a bounded number of crossings per edge. We then prove that graphs with maximum degree $\Delta$ that exclude a fixed minor are 3-colourable with clustering $O(\Delta^5)$ . The best previous bound for this result was exponential in $\Delta$ .

scholarly journals The Complexity of Approximating the Matching Polynomial in the Complex Plane

2021 ◽  
Vol 13 (2) ◽  
pp. 1-37
Author(s):  
Ivona Bezáková ◽  
Andreas Galanis ◽  
Leslie Ann Goldberg ◽  
Daniel Štefankovič
Keyword(s):  
Real Axis ◽  
Complex Plane ◽  
Maximum Degree ◽  
Negative Real Axis ◽  
Positive Real ◽  
Bounded Degree ◽  
Matching Polynomial ◽  
Correlation Decay ◽  
Connective Constant ◽  
General Complex

We study the problem of approximating the value of the matching polynomial on graphs with edge parameter γ, where γ takes arbitrary values in the complex plane. When γ is a positive real, Jerrum and Sinclair showed that the problem admits an FPRAS on general graphs. For general complex values of γ, Patel and Regts, building on methods developed by Barvinok, showed that the problem admits an FPTAS on graphs of maximum degree Δ as long as γ is not a negative real number less than or equal to −1/(4(Δ −1)). Our first main result completes the picture for the approximability of the matching polynomial on bounded degree graphs. We show that for all Δ ≥ 3 and all real γ less than −1/(4(Δ −1)), the problem of approximating the value of the matching polynomial on graphs of maximum degree Δ with edge parameter γ is #P-hard. We then explore whether the maximum degree parameter can be replaced by the connective constant. Sinclair et al. showed that for positive real γ, it is possible to approximate the value of the matching polynomial using a correlation decay algorithm on graphs with bounded connective constant (and potentially unbounded maximum degree). We first show that this result does not extend in general in the complex plane; in particular, the problem is #P-hard on graphs with bounded connective constant for a dense set of γ values on the negative real axis. Nevertheless, we show that the result does extend for any complex value γ that does not lie on the negative real axis. Our analysis accounts for complex values of γ using geodesic distances in the complex plane in the metric defined by an appropriate density function.

scholarly journals The distinguishing number of Cartesian products of complete graphs

2008 ◽  
Vol 29 (4) ◽  
pp. 922-929 ◽  
Author(s):  
Wilfried Imrich ◽  
Janja Jerebic ◽  
Sandi Klavžar
Keyword(s):  
Complete Graphs ◽  
Cartesian Products ◽  
Distinguishing Number

scholarly journals On randomly 3-axial graphs

1982 ◽  
Vol 25 (2) ◽  
pp. 187-206
Author(s):  
Yousef Alavi ◽  
Sabra S. Anderson ◽  
Gary Chartrand ◽  
S.F. Kapoor
Keyword(s):  
Bipartite Graphs ◽  
Disjoint Paths ◽  
Complete Graphs ◽  
Initial Vertex ◽  
Complete Bipartite Graphs

A graph G, every vertex of which has degree at least three, is randomly 3-axial if for each vertex v of G, any ordered collection of three paths in G of length one with initial vertex v can be cyclically randomly extended to produce three internally disjoint paths which contain all the vertices of G. Randomly 3-axial graphs of order p > 4 are characterized for p ≢ 1 (mod 3), and are shown to be either complete graphs or certain regular complete bipartite graphs.

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