The Upper Bound on the Eulerian Recurrent Lengths of Complete Graphs Obtained by an IP Solver

2018 ◽  
pp. 199-208
Author(s):  
Shuji Jimbo ◽  
Akira Maruoka
Keyword(s):  
Upper Bound ◽  
Complete Graphs

scholarly journals On Partitioning and Packing Products with Rectangles

1994 ◽  
Vol 3 (4) ◽  
pp. 429-434 ◽  
Author(s):  
Rudolf Ahlswede ◽  
Ning Cai
Keyword(s):  
Upper Bound ◽  
Complete Graphs ◽  
Minimal Size ◽  
Uniform Hypergraphs ◽  
Image Position ◽  
Special Case

In [1] we introduced and studied for product hypergraphs where ℋi = (i,ℰi), the minimal size π(ℋn) of a partition of into sets that are elements of . The main result was thatif the ℋis are graphs with all loops included. A key step in the proof concerns the special case of complete graphs. Here we show that (1) also holds when the ℋi are complete d-uniform hypergraphs with all loops included, subject to a condition on the sizes of the i. We also present an upper bound on packing numbers.

scholarly journals On the Widom–Rowlinson Occupancy Fraction in Regular Graphs

2016 ◽  
Vol 26 (2) ◽  
pp. 183-194 ◽  
Author(s):  
EMMA COHEN ◽  
WILL PERKINS ◽  
PRASAD TETALI
Keyword(s):  
Partition Function ◽  
Regular Graph ◽  
Upper Bound ◽  
Complete Graphs ◽  
Regular Graphs ◽  
Interacting Particles ◽  
Special Case ◽  
Random Configuration

We consider the Widom–Rowlinson model of two types of interacting particles on d-regular graphs. We prove a tight upper bound on the occupancy fraction, the expected fraction of vertices occupied by a particle under a random configuration from the model. The upper bound is achieved uniquely by unions of complete graphs on d + 1 vertices, Kd+1. As a corollary we find that Kd+1 also maximizes the normalized partition function of the Widom–Rowlinson model over the class of d-regular graphs. A special case of this shows that the normalized number of homomorphisms from any d-regular graph G to the graph HWR, a path on three vertices with a loop on each vertex, is maximized by Kd+1. This proves a conjecture of Galvin.

scholarly journals Binomial Edge Ideals of Graphs

10.37236/2349 ◽  
2012 ◽  
Vol 19 (2) ◽  
Author(s):  
Dariush Kiani ◽  
Sara Saeedi
Keyword(s):  
Upper Bound ◽  
Betti Number ◽  
Complete Graphs ◽  
Edge Ideal ◽  
Closed Graph ◽  
Edge Ideals ◽  
Linear Resolution

We characterize all graphs whose binomial edge ideals have a linear resolution. Indeed, we show that complete graphs are the only graphs with this property. We also compute some graded components of the first Betti number of the binomial edge ideal of a graph with respect to the graphical terms. Finally, we give an upper bound for the Castelnuovo-Mumford regularity of the binomial edge ideal of a closed graph.

scholarly journals Randomized Rumour Spreading: The Effect of the Network Topology

2014 ◽  
Vol 24 (2) ◽  
pp. 457-479 ◽  
Author(s):  
KONSTANTINOS PANAGIOTOU ◽  
XAVIER PÉREZ-GIMÉNEZ ◽  
THOMAS SAUERWALD ◽  
HE SUN
Keyword(s):  
Random Graph ◽  
Network Topology ◽  
Upper Bound ◽  
High Probability ◽  
Spectral Expansion ◽  
Complete Graphs ◽  
General Graph ◽  
Graph Class ◽  
Network Topologies ◽  
Push Model

We consider the popular and well-studied push model, which is used to spread information in a given network with n vertices. Initially, some vertex owns a rumour and passes it to one of its neighbours, which is chosen randomly. In each of the succeeding rounds, every vertex that knows the rumour informs a random neighbour. It has been shown on various network topologies that this algorithm succeeds in spreading the rumour within O(log n) rounds. However, many studies are quite coarse and involve huge constants that do not allow for a direct comparison between different network topologies. In this paper, we analyse the push model on several important families of graphs, and obtain tight runtime estimates. We first show that, for any almost-regular graph on n vertices with small spectral expansion, rumour spreading completes after log2n + log n+o(log n) rounds with high probability. This is the first result that exhibits a general graph class for which rumour spreading is essentially as fast as on complete graphs. Moreover, for the random graph G(n,p) with p=c log n/n, where c > 1, we determine the runtime of rumour spreading to be log2n + γ (c)log n with high probability, where γ(c) = clog(c/(c−1)). In particular, this shows that the assumption of almost regularity in our first result is necessary. Finally, for a hypercube on n=2d vertices, the runtime is with high probability at least (1+β) ⋅ (log2n + log n), where β > 0. This reveals that the push model on hypercubes is slower than on complete graphs, and thus shows that the assumption of small spectral expansion in our first result is also necessary. In addition, our results combined with the upper bound of O(log n) for the hypercube (see [11]) imply that the push model is faster on hypercubes than on a random graph G(n, clog n/n), where c is sufficiently close to 1.

scholarly journals An improved upper bound on the maximum degree of terminal-pairable complete graphs

2018 ◽  
Vol 341 (9) ◽  
pp. 2606-2607 ◽  
Author(s):  
António Girão ◽  
Gábor Mészáros
Keyword(s):  
Upper Bound ◽  
Maximum Degree ◽  
Complete Graphs

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 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 Product Dimension of Forests and Bounded Treewidth Graphs

10.37236/2698 ◽  
2013 ◽  
Vol 20 (3) ◽  
Author(s):  
L. Sunil Chandran ◽  
Rogers Mathew ◽  
Deepak Rajendraprasad ◽  
Roohani Sharma
Keyword(s):  
Natural Number ◽  
Direct Product ◽  
Upper Bound ◽  
Latin Squares ◽  
Complete Graphs ◽  
Induced Subgraph ◽  
Orthogonal Latin Squares ◽  
Bounded Treewidth ◽  
Product Dimension

The product dimension of a graph $G$ is defined as the minimum natural number $l$ such that $G$ is an induced subgraph of a direct product of $l$ complete graphs. In this paper we study the product dimension of forests, bounded treewidth graphs and $k$-degenerate graphs. We show that every forest on $n$ vertices has product dimension at most $1.441 \log n + 3$. This improves the best known upper bound of $3 \log n$ for the same due to Poljak and Pultr. The technique used in arriving at the above bound is extended and combined with a well-known result on the existence of orthogonal Latin squares to show that every graph on $n$ vertices with treewidth at most $t$ has product dimension at most $(t+2)(\log n + 1)$. We also show that every $k$-degenerate graph on $n$ vertices has product dimension at most $\lceil 5.545 k \log n \rceil + 1$. This improves the upper bound of  $32 k \log n$ for the same by Eaton and  Rődl.

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