scholarly journals Shannon Capacity and the Categorical Product

10.37236/9113 ◽  
2021 ◽  
Vol 28 (1) ◽  
Author(s):  
Gábor Simonyi
Keyword(s):  
Lower Bound ◽  
Recent Result ◽  
Chromatic Number ◽  
Upper Bound ◽  
Graph Invariant ◽  
Shannon Capacity ◽  
Graph Homomorphisms ◽  
Complementary Graph ◽  
Product Of Graphs ◽  
Monotone Graph

Shannon OR-capacity $C_{\rm OR}(G)$ of a graph $G$, that is the traditionally more often used Shannon AND-capacity of the complementary graph, is a homomorphism monotone graph parameter therefore $C_{\rm OR}(F\times G)\leqslant\min\{C_{\rm OR}(F),C_{\rm OR}(G)\}$ holds for every pair of graphs, where $F\times G$ is the categorical product of graphs $F$ and $G$. Here we initiate the study of the question when could we expect equality in this inequality. Using a strong recent result of Zuiddam, we show that if this "Hedetniemi-type" equality is not satisfied for some pair of graphs then the analogous equality is also not satisfied for this graph pair by some other graph invariant that has a much "nicer" behavior concerning some different graph operations. In particular, unlike Shannon OR-capacity or the chromatic number, this other invariant is both multiplicative under the OR-product and additive under the join operation, while it is also nondecreasing along graph homomorphisms. We also present a natural lower bound on $C_{\rm OR}(F\times G)$ and elaborate on the question of how to find graph pairs for which it is known to be strictly less than the upper bound $\min\{C_{\rm OR}(F),C_{\rm OR}(G)\}$. We present such graph pairs using the properties of Paley graphs.

scholarly journals On r-dynamic coloring of comb graphs

2021 ◽  
Vol 27 (2) ◽  
pp. 191-200
Author(s):  
K. Kalaiselvi ◽  
◽  
N. Mohanapriya ◽  
J. Vernold Vivin ◽  
◽  
...  
Keyword(s):  
Lower Bound ◽  
Chromatic Number ◽  
Upper Bound ◽  
Line Graph ◽  
Proper Coloring ◽  
Total Graph ◽  
Middle Graph ◽  
Different Color

An r-dynamic coloring of a graph G is a proper coloring of G such that every vertex in V(G) has neighbors in at least $\min\{d(v),r\}$ different color classes. The r-dynamic chromatic number of graph G denoted as $\chi_r (G)$, is the least k such that G has a coloring. In this paper we obtain the r-dynamic chromatic number of the central graph, middle graph, total graph, line graph, para-line graph and sub-division graph of the comb graph $P_n\odot K_1$ denoted by $C(P_n\odot K_1), M(P_n\odot K_1), T(P_n\odot K_1), L(P_n\odot K_1), P(P_n\odot K_1)$ and $S(P_n\odot K_1)$ respectively by finding the upper bound and lower bound for the r-dynamic chromatic number of the Comb graph.

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.

scholarly journals Off-Diagonal Ordered Ramsey Numbers of Matchings

10.37236/8085 ◽  
2019 ◽  
Vol 26 (2) ◽  
Author(s):  
Dhruv Rohatgi
Keyword(s):  
Lower Bound ◽  
Chromatic Number ◽  
Upper Bound ◽  
Edge Coloring ◽  
Ramsey Number ◽  
Relative Order ◽  
Ramsey Numbers ◽  
Asymptotic Bounds ◽  
Special Cases ◽  
Ordered Graphs

For ordered graphs $G$ and $H$, the ordered Ramsey number $r_<(G,H)$ is the smallest $n$ such that every red/blue edge coloring of the complete ordered graph on vertices $\{1,\dots,n\}$ contains either a blue copy of $G$ or a red copy of $H$, where the embedding must preserve the relative order of vertices. One number of interest, first studied by Conlon, Fox, Lee, and Sudakov, is the off-diagonal ordered Ramsey number $r_<(M, K_3)$, where $M$ is an ordered matching on $n$ vertices. In particular, Conlon et al. asked what asymptotic bounds (in $n$) can be obtained for $\max r_<(M, K_3)$, where the maximum is over all ordered matchings $M$ on $n$ vertices. The best-known upper bound is $O(n^2/\log n)$, whereas the best-known lower bound is $\Omega((n/\log n)^{4/3})$, and Conlon et al. hypothesize that there is some fixed $\epsilon > 0$ such that $r_<(M, K_3) = O(n^{2-\epsilon})$ for every ordered matching $M$. We resolve two special cases of this conjecture. We show that the off-diagonal ordered Ramsey numbers for ordered matchings in which edges do not cross are nearly linear. We also prove a truly sub-quadratic upper bound for random ordered matchings with interval chromatic number $2$.

scholarly journals Improved Bounds for Radiok-Chromatic Number of HypercubeQn

2011 ◽  
Vol 2011 ◽  
pp. 1-7
Author(s):  
Laxman Saha ◽  
Pratima Panigrahi ◽  
Pawan Kumar
Keyword(s):  
Lower Bound ◽  
Chromatic Number ◽  
Graph Coloring ◽  
Upper Bound ◽  
Channel Assignment ◽  
Assignment Problem ◽  
Communication Problem ◽  
Channel Assignment Problem

A number of graph coloring problems have their roots in a communication problem known as the channel assignment problem. The channel assignment problem is the problem of assigning channels (nonnegative integers) to the stations in an optimal way such that interference is avoided as reported by Hale (2005). Radiok-coloring of a graph is a special type of channel assignment problem. Kchikech et al. (2005) have given a lower and an upper bound for radiok-chromatic number of hypercubeQn, and an improvement of their lower bound was obtained by Kola and Panigrahi (2010). In this paper, we further improve Kola et al.'s lower bound as well as Kchikeck et al.'s upper bound. Also, our bounds agree for nearly antipodal number ofQnwhenn≡2(mod 4).

scholarly journals The Chromatic Number and the Least Eigenvalue of a Graph

10.37236/2043 ◽  
2012 ◽  
Vol 19 (1) ◽  
Author(s):  
Yi-Zheng Fan ◽  
Gui-Dong Yu ◽  
Yi Wang
Keyword(s):  
Lower Bound ◽  
Chromatic Number ◽  
Structural Property ◽  
Upper Bound ◽  
Least Eigenvalue ◽  
Fixed Order

In this paper we get a structural property for a graph having the minimal least eigenvalue among all graphs of fixed order and given chromatic number, and characterize such graphs under the condition that the chromatic number is not larger than half the order of the graph.  As a result, we obtain a lower bound on the least eigenvalue in terms of the chromatic number, and an upper bound on the chromatic number in terms of the least eigenvalue of a graph.

r-Dynamic coloring of the corona product of graphs

2020 ◽  
Vol 12 (02) ◽  
pp. 2050019
Author(s):  
Arika Indah Kristiana ◽  
M. Imam Utoyo ◽  
Ridho Alfarisi ◽  
Dafik
Keyword(s):  
Lower Bound ◽  
Chromatic Number ◽  
Product Of Graphs ◽  
Corona Product

Let [Formula: see text] be a graph. A proper [Formula: see text]-coloring of graph [Formula: see text] is [Formula: see text]-dynamic coloring if for every [Formula: see text], the neighbors of vertex [Formula: see text] receive at least min[Formula: see text] different colors. The minimum [Formula: see text] such that graph [Formula: see text] has [Formula: see text]-dynamic [Formula: see text] coloring is called the [Formula: see text]-dynamic chromatic number, denoted by [Formula: see text]. In this paper, we study the [Formula: see text]-dynamic coloring of corona product of graph. The corona product of graph is obtained by taking a number of vertices [Formula: see text] copy of [Formula: see text], and making the [Formula: see text]th of [Formula: see text] adjacent to every vertex of the [Formula: see text]th copy of [Formula: see text]. We obtain the lower bound of [Formula: see text]-dynamic chromatic number of corona product of graphs and some exact value.

scholarly journals An Exponential Separation Between MA and AM Proofs of Proximity

2021 ◽  
Vol 30 (2) ◽  
Author(s):  
Tom Gur ◽  
Yang P. Liu ◽  
Ron D. Rothblum
Keyword(s):  
Quantum Information ◽  
Lower Bound ◽  
Recent Result ◽  
Query Complexity ◽  
Random String ◽  
Natural Property ◽  
Public And Private ◽  
Alternate Proof ◽  
Interactive Proofs ◽  
Exponential Separation

AbstractInteractive proofs of proximity allow a sublinear-time verifier to check that a given input is close to the language, using a small amount of communication with a powerful (but untrusted) prover. In this work, we consider two natural minimally interactive variants of such proofs systems, in which the prover only sends a single message, referred to as the proof. The first variant, known as -proofs of Proximity (), is fully non-interactive, meaning that the proof is a function of the input only. The second variant, known as -proofs of Proximity (), allows the proof to additionally depend on the verifier's (entire) random string. The complexity of both s and s is the total number of bits that the verifier observes—namely, the sum of the proof length and query complexity. Our main result is an exponential separation between the power of s and s. Specifically, we exhibit an explicit and natural property $$\Pi$$ Π that admits an with complexity $$O(\log n)$$ O ( log n ) , whereas any for $$\Pi$$ Π has complexity $$\tilde{\Omega}(n^{1/4})$$ Ω ~ ( n 1 / 4 ) , where n denotes the length of the input in bits. Our lower bound also yields an alternate proof, which is more general and arguably much simpler, for a recent result of Fischer et al. (ITCS, 2014). Also, Aaronson (Quantum Information & Computation 2012) has shown a $$\Omega(n^{1/6})$$ Ω ( n 1 / 6 ) lower bound for the same property $$\Pi$$ Π .Lastly, we also consider the notion of oblivious proofs of proximity, in which the verifier's queries are oblivious to the proof. In this setting, we show that s can only be quadratically stronger than s. As an application of this result, we show an exponential separation between the power of public and private coin for oblivious interactive proofs of proximity.

scholarly journals On the Interactive Capacity of Finite-State Protocols

Entropy ◽  
2020 ◽  
Vol 23 (1) ◽  
pp. 17
Author(s):  
Assaf Ben-Yishai ◽  
Young-Han Kim ◽  
Rotem Oshman ◽  
Ofer Shayevitz
Keyword(s):  
Lower Bounds ◽  
Upper Bound ◽  
Noisy Channel ◽  
Shannon Capacity ◽  
Finite State

The interactive capacity of a noisy channel is the highest possible rate at which arbitrary interactive protocols can be simulated reliably over the channel. Determining the interactive capacity is notoriously difficult, and the best known lower bounds are far below the associated Shannon capacity, which serves as a trivial (and also generally the best known) upper bound. This paper considers the more restricted setup of simulating finite-state protocols. It is shown that all two-state protocols, as well as rich families of arbitrary finite-state protocols, can be simulated at the Shannon capacity, establishing the interactive capacity for those families of protocols.

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