scholarly journals A Lower Bound on the Average Degree Forcing a Minor

10.37236/8847 ◽  
2020 ◽  
Vol 27 (1) ◽  
Author(s):  
Sergey Norin ◽  
Bruce Reed ◽  
Andrew Thomason ◽  
David R. Wood
Keyword(s):  
Lower Bound ◽  
Upper Bound ◽  
Constant Factor ◽  
Average Degree ◽  
Complete Graphs ◽  
Sparse Graphs ◽  
Dense Graphs ◽  
A Minor

We show that for sufficiently large $d$ and for $t\geq d+1$,  there is a graph $G$ with average degree $(1-\varepsilon)\lambda  t \sqrt{\ln d}$ such that almost every graph $H$ with $t$ vertices and average degree $d$ is not a minor of $G$, where $\lambda=0.63817\dots$ is an explicitly defined constant. This generalises analogous results for complete graphs by Thomason (2001) and for general dense graphs by Myers and Thomason (2005). It also shows that an upper bound for sparse graphs by Reed and Wood (2016) is best possible up to a constant factor.

scholarly journals The Degree-Diameter Problem for Sparse Graph Classes

10.37236/4313 ◽  
2015 ◽  
Vol 22 (2) ◽  
Author(s):  
Guillermo Pineda-Villavicencio ◽  
David R. Wood
Keyword(s):  
Maximum Degree ◽  
Constant Factor ◽  
Average Degree ◽  
Sparse Graph ◽  
Sparse Graphs ◽  
Graph Classes ◽  
Minor Free Graphs

The degree-diameter problem asks for the maximum number of vertices in a graph with maximum degree $\Delta$ and diameter $k$. For fixed $k$, the answer is $\Theta(\Delta^k)$. We consider the degree-diameter problem for particular classes of sparse graphs, and establish the following results. For graphs of bounded average degree the answer is $\Theta(\Delta^{k-1})$, and for graphs of bounded arboricity the answer is $\Theta(\Delta^{\lfloor k/2\rfloor})$, in both cases for fixed $k$. For graphs of given treewidth, we determine the maximum number of vertices up to a constant factor. Other precise bounds are given for graphs embeddable on a given surface and apex-minor-free graphs.

scholarly journals Induced Ramsey Number for a Star Versus a Fixed Graph

10.37236/9358 ◽  
2021 ◽  
Vol 28 (1) ◽  
Author(s):  
Maria Axenovich ◽  
Izolda Gorgol
Keyword(s):  
Lower Bound ◽  
Chromatic Number ◽  
Upper Bound ◽  
Constant Factor ◽  
Ramsey Number ◽  
Large N ◽  
Star Versus

We write $F{\buildrel {\text{ind}} \over \longrightarrow}(H,G)$ for graphs $F, G,$ and $H$, if for any coloring of the edges of $F$ in red and blue, there is either a red induced copy of $H$ or a blue induced copy of $G$. For graphs $G$ and $H$, let $\mathrm{IR}(H,G)$ be the smallest number of vertices in a graph $F$ such that $F{\buildrel {\text{ind}} \over \longrightarrow}(H,G)$. In this note we consider the case when $G$ is a star on $n$ edges, for large $n$ and $H$ is a fixed graph. We prove that  $$ (\chi(H)-1) n \leq \mathrm{IR}(H, K_{1,n}) \leq (\chi(H)-1)^2n + \epsilon n,$$ for any $\epsilon>0$,  sufficiently large $n$, and $\chi(H)$ denoting the chromatic number of $H$. The lower bound is asymptotically tight  for any fixed bipartite $H$. The upper bound is attained up to a constant factor, for example when $H$ is a clique.

State Complexity of k-Union and k-Intersection for Prefix-Free Regular Languages

2015 ◽  
Vol 26 (02) ◽  
pp. 211-227 ◽  
Author(s):  
Hae-Sung Eom ◽  
Yo-Sub Han ◽  
Kai Salomaa
Keyword(s):  
Lower Bound ◽  
Upper Bound ◽  
Finite Automata ◽  
Constant Factor ◽  
The State ◽  
Regular Languages ◽  
Deterministic Finite Automata ◽  
State Complexity ◽  
Binary Alphabet ◽  
Multiple Intersections

We investigate the state complexity of multiple unions and of multiple intersections for prefix-free regular languages. Prefix-free deterministic finite automata have their own unique structural properties that are crucial for obtaining state complexity upper bounds that are improved from those for general regular languages. We present a tight lower bound construction for k-union using an alphabet of size k + 1 and for k-intersection using a binary alphabet. We prove that the state complexity upper bound for k-union cannot be reached by languages over an alphabet with less than k symbols. We also give a lower bound construction for k-union using a binary alphabet that is within a constant factor of the upper bound.

FITTING FLATS TO POINTS WITH OUTLIERS

2011 ◽  
Vol 21 (05) ◽  
pp. 559-569
Author(s):  
GUILHERME D. DA FONSECA
Keyword(s):  
Lower Bound ◽  
Computer Science ◽  
Upper Bound ◽  
Arbitrary Constant ◽  
Fundamental Problem ◽  
Constant Factor ◽  
Infinite Cylinder ◽  
Running Time ◽  
Point Set ◽  
Set Of Points

Determining the best shape to fit a set of points is a fundamental problem in many areas of computer science. We present an algorithm to approximate the k-flat that best fits a set of n points with n - m outliers. This problem generalizes the smallest m-enclosing ball, infinite cylinder, and slab. Our algorithm gives an arbitrary constant factor approximation in O(nk+2/m) time, regardless of the dimension of the point set. While our upper bound nearly matches the lower bound, the algorithm may not be feasible for large values of k. Fortunately, for some practical sets of inliers, we reduce the running time to O(nk+2/mk+1), which is linear when m = Ω(n).

scholarly journals An Exact Performance Bound for an ${O (m+n)}$ Time Greedy Matching Procedure

10.37236/1310 ◽  
1997 ◽  
Vol 4 (1) ◽  
Author(s):  
Andrew Shapira
Keyword(s):  
Lower Bound ◽  
Extremal Function ◽  
Hybrid Algorithms ◽  
Maximum Matching ◽  
Performance Bound ◽  
Sparse Graphs ◽  
Augmenting Path ◽  
Dense Graphs ◽  
Matching Procedure

We prove an exact lower bound on $\gamma(G)$, the size of the smallest matching that a certain $O(m+n)$ time greedy matching procedure may find for a given graph $G$ with $n$ vertices and $m$ edges. The bound is precisely Erdős and Gallai's extremal function that gives the size of the smallest maximum matching, over all graphs with $n$ vertices and $m$ edges. Thus the greedy procedure is optimal in the sense that when only $n$ and $m$ are specified, no algorithm can be guaranteed to find a larger matching than the greedy procedure. The greedy procedure and augmenting path algorithms are seen to be complementary: the greedy procedure finds a large matching for dense graphs, while augmenting path algorithms are fast for sparse graphs. Well known hybrid algorithms consisting of the greedy procedure followed by an augmenting path algorithm are shown to be faster than the augmenting path algorithm alone. The lower bound on $\gamma(G)$ is a stronger version of Erdős and Gallai's result, and so the proof of the lower bound is a new way of proving of Erdős and Gallai's result.

IMPROVED BOUNDS FOR THE CROSSING NUMBER OF THE MESH OF TREES

2003 ◽  
Vol 04 (01) ◽  
pp. 17-35 ◽  
Author(s):  
ROBERT CIMIKOWSKI ◽  
IMRICH VRT'O
Keyword(s):  
Lower Bound ◽  
Upper Bound ◽  
Crossing Number ◽  
Constant Factor ◽  
Mesh Of Trees

Improved bounds for the crossing number of the mesh of trees graph, Mn, are derived. In particular, we derive a new lower bound of [Formula: see text] which improves on the previous bound of Leighton [11] by a constant factor, and an upper bound of [Formula: see text]. In addition, we construct drawings of Mn which achieve the upper bound number of crossings. We also prove that the crossing number of M4 is 4.

scholarly journals On the Maximal Colorings of Complete Graphs Without Some Small Properly Colored Subgraphs

2021 ◽  
Author(s):  
Chunqiu Fang ◽  
Ervin Győri ◽  
Jimeng Xiao
Keyword(s):  
Lower Bound ◽  
Upper Bound ◽  
Edge Coloring ◽  
Complete Graphs

AbstractLet $$\mathrm{pr}(K_{n}, G)$$ pr ( K n , G ) be the maximum number of colors in an edge-coloring of $$K_{n}$$ K n with no properly colored copy of G. For a family $${\mathcal {F}}$$ F of graphs, let $$\mathrm{ex}(n, {\mathcal {F}})$$ ex ( n , F ) be the maximum number of edges in a graph G on n vertices which does not contain any graphs in $${\mathcal {F}}$$ F as subgraphs. In this paper, we show that $$\mathrm{pr}(K_{n}, G)-\mathrm{ex}(n, \mathcal {G'})=o(n^{2}), $$ pr ( K n , G ) - ex ( n , G ′ ) = o ( n 2 ) , where $$\mathcal {G'}=\{G-M: M \text { is a matching of }G\}$$ G ′ = { G - M : M is a matching of G } . Furthermore, we determine the value of $$\mathrm{pr}(K_{n}, P_{l})$$ pr ( K n , P l ) for sufficiently large n and the exact value of $$\mathrm{pr}(K_{n}, G)$$ pr ( K n , G ) , where G is $$C_{5}, C_{6}$$ C 5 , C 6 and $$K_{4}^{-}$$ K 4 - , respectively. Also, we give an upper bound and a lower bound of $$\mathrm{pr}(K_{n}, K_{2,3})$$ pr ( K n , K 2 , 3 ) .

scholarly journals The Random Graph Intuition for the Tournament Game

2015 ◽  
Vol 25 (1) ◽  
pp. 76-88 ◽  
Author(s):  
DENNIS CLEMENS ◽  
HEIDI GEBAUER ◽  
ANITA LIEBENAU
Keyword(s):  
Lower Bound ◽  
Random Graph ◽  
Complete Graph ◽  
Upper Bound ◽  
Winning Strategy ◽  
Constant Factor ◽  
Additive Constant ◽  
Underlying Graph ◽  
Straightforward Application

In the tournament game two players, called Maker and Breaker, alternately take turns in claiming an unclaimed edge of the complete graph Kn and selecting one of the two possible orientations. Before the game starts, Breaker fixes an arbitrary tournament Tk on k vertices. Maker wins if, at the end of the game, her digraph contains a copy of Tk; otherwise Breaker wins. In our main result, we show that Maker has a winning strategy for k = (2 − o(1))log2n, improving the constant factor in previous results of Beck and the second author. This is asymptotically tight since it is known that for k = (2 − o(1))log2n Breaker can prevent the underlying graph of Maker's digraph from containing a k-clique. Moreover, the precise value of our lower bound differs from the upper bound only by an additive constant of 12.We also discuss the question of whether the random graph intuition, which suggests that the threshold for k is asymptotically the same for the game played by two ‘clever’ players and the game played by two ‘random’ players, is supported by the tournament game. It will turn out that, while a straightforward application of this intuition fails, a more subtle version of it is still valid.Finally, we consider the orientation game version of the tournament game, where Maker wins the game if the final digraph – also containing the edges directed by Breaker – possesses a copy of Tk. We prove that in that game Breaker has a winning strategy for k = (4 + o(1))log2n.

Injective edge coloring of sparse graphs

2018 ◽  
Vol 10 (02) ◽  
pp. 1850022
Author(s):  
Yuehua Bu ◽  
Chentao Qi
Keyword(s):  
Upper Bound ◽  
Edge Coloring ◽  
Average Degree ◽  
Sparse Graphs ◽  
Maximum Average Degree ◽  
Coloring Number

A [Formula: see text]-injective edge coloring of a graph [Formula: see text] is a coloring [Formula: see text], such that if [Formula: see text], [Formula: see text] and [Formula: see text] are consecutive edges in [Formula: see text], then [Formula: see text]. [Formula: see text] has a [Formula: see text]-injective edge coloring[Formula: see text] is called the injective edge coloring number. In this paper, we consider the upper bound of [Formula: see text] in terms of the maximum average degree mad[Formula: see text], where [Formula: see text].

scholarly journals Average Degree Conditions Forcing a Minor

10.37236/5321 ◽  
2016 ◽  
Vol 23 (1) ◽  
Author(s):  
Daniel J. Harvey ◽  
David R. Wood
Keyword(s):  
Recent Result ◽  
Complete Graph ◽  
Open Problem ◽  
Complete Bipartite Graph ◽  
Constant Factor ◽  
Average Degree ◽  
Sparse Graph ◽  
Arbitrary Graph ◽  
A Minor ◽  
High Degree

Mader first proved that high average degree forces a given graph as a minor. Often motivated by Hadwiger's Conjecture, much research has focused on the average degree required to force a complete graph as a minor. Subsequently, various authors have considered the average degree required to force an arbitrary graph $H$ as a minor. Here, we strengthen (under certain conditions) a recent result by Reed and Wood, giving better bounds on the average degree required to force an $H$-minor when $H$ is a sparse graph with many high degree vertices. This solves an open problem of Reed and Wood, and also generalises (to within a constant factor) known results when $H$ is an unbalanced complete bipartite graph.

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