scholarly journals New Upper Bound for Sums of Dilates

10.37236/6576 ◽  
2017 ◽  
Vol 24 (3) ◽  
Author(s):  
Albert Bush ◽  
Yi Zhao
Keyword(s):  
Upper Bound ◽  
Bipartite Graphs ◽  
Binary Expansion ◽  
Main Technique

For $\lambda \in \mathbb{Z}$, let $\lambda \cdot A = \{ \lambda a : a \in A\}$. Suppose $r, h\in \mathbb{Z}$ are sufficiently large and comparable to each other. We prove that if $|A+A| \le K |A|$ and $\lambda_1, \ldots, \lambda_h \le 2^r$, then \[ |\lambda_1 \cdot A + \ldots + \lambda_h \cdot A | \le K^{ 7 rh /\ln (r+h) } |A|. \]This improves upon a result of Bukh who shows that\[ |\lambda_1 \cdot A + \ldots + \lambda_h \cdot A | \le K^{O(rh)} |A|. \]Our main technique is to combine Bukh's idea of considering the binary expansion of $\lambda_i$ with a result on biclique decompositions of bipartite graphs.

scholarly journals On the Strong Equitable Vertex 2-Arboricity of Complete Bipartite Graphs

Mathematics ◽  
2020 ◽  
Vol 8 (10) ◽  
pp. 1778
Author(s):  
Fangyun Tao ◽  
Ting Jin ◽  
Yiyou Tu
Keyword(s):  
Upper Bound ◽  
Maximum Degree ◽  
Bipartite Graphs ◽  
Equitable Partition ◽  
Special Cases ◽  
Vertex Set ◽  
Complete Bipartite Graphs

An equitable partition of a graph G is a partition of the vertex set of G such that the sizes of any two parts differ by at most one. The strong equitable vertexk-arboricity of G, denoted by vak≡(G), is the smallest integer t such that G can be equitably partitioned into t′ induced forests for every t′≥t, where the maximum degree of each induced forest is at most k. In this paper, we provide a general upper bound for va2≡(Kn,n). Exact values are obtained in some special cases.

scholarly journals Abstract Tropical Linear Programming

10.37236/7718 ◽  
2020 ◽  
Vol 27 (2) ◽  
Author(s):  
Georg Loho
Keyword(s):  
Linear Programming ◽  
Upper Bound ◽  
Axiomatic Approach ◽  
Bipartite Graphs ◽  
Complexity Measure ◽  
Integer Vector ◽  
Feasible Point ◽  
Additional Sign

In this paper we develop a combinatorial abstraction of tropical linear programming. This generalizes the search for a feasible point of a system of min-plus-inequalities. We obtain an algorithm based on an axiomatic approach to this generalization.  It builds on the introduction of signed tropical matroids based on the polyhedral properties of triangulations of the product of two simplices and the combinatorics of the associated set of bipartite graphs with an additional sign information. Finally, we establish an upper bound for our feasibility algorithm applied to a system of min-plus-inequalities in terms of the secondary fan of a product of two simplices. The appropriate complexity measure is a shortest integer vector in a cone of the secondary fan associated to the system.

scholarly journals The Number of Independent Sets in a Regular Graph

2009 ◽  
Vol 19 (2) ◽  
pp. 315-320 ◽  
Author(s):  
YUFEI ZHAO
Keyword(s):  
Regular Graph ◽  
Upper Bound ◽  
Disjoint Union ◽  
Short Proof ◽  
Independent Sets ◽  
Bipartite Graphs ◽  
Independence Polynomial ◽  
Bipartite Case

We show that the number of independent sets in an N-vertex, d-regular graph is at most (2d+1 − 1)N/2d, where the bound is sharp for a disjoint union of complete d-regular bipartite graphs. This settles a conjecture of Alon in 1991 and Kahn in 2001. Kahn proved the bound when the graph is assumed to be bipartite. We give a short proof that reduces the general case to the bipartite case. Our method also works for a weighted generalization, i.e., an upper bound for the independence polynomial of a regular graph.

Total edge–vertex domination

2020 ◽  
Vol 54 ◽  
pp. 1 ◽  
Author(s):  
Abdulgani Sahin ◽  
Bünyamin Sahin
Keyword(s):  
Upper Bound ◽  
Dominating Set ◽  
Domination Number ◽  
Bipartite Graphs ◽  
Np Hard ◽  
Minimum Cardinality

An edge e ev-dominates a vertex v which is a vertex of e, as well as every vertex adjacent to v. A subset D ⊆ E is an edge-vertex dominating set (in simply, ev-dominating set) of G, if every vertex of a graph G is ev-dominated by at least one edge of D. The minimum cardinality of an ev-dominating set is named with ev-domination number and denoted by γev(G). A subset D ⊆ E is a total edge-vertex dominating set (in simply, total ev-dominating set) of G, if D is an ev-dominating set and every edge of D shares an endpoint with other edge of D. The total ev-domination number of a graph G is denoted with γevt(G) and it is equal to the minimum cardinality of a total ev-dominating set. In this paper, we initiate to study total edge-vertex domination. We first show that calculating the number γevt(G) for bipartite graphs is NP-hard. We also show the upper bound γevt(T) ≤ (n − l + 2s − 1)∕2 for the total ev-domination number of a tree T, where T has order n, l leaves and s support vertices and we characterize the trees achieving this upper bound. Finally, we obtain total ev-domination number of paths and cycles.

On Turán exponents of bipartite graphs

2021 ◽  
pp. 1-12
Author(s):  
Tao Jiang ◽  
Jie Ma ◽  
Liana Yepremyan
Keyword(s):  
Bipartite Graph ◽  
Rational Number ◽  
Upper Bound ◽  
Bipartite Graphs ◽  
Affirmative Answer ◽  
Finite Family ◽  
Turán Number ◽  
New Form ◽  
Single Graph

Abstract A long-standing conjecture of Erdős and Simonovits asserts that for every rational number $r\in (1,2)$ there exists a bipartite graph H such that $\mathrm{ex}(n,H)=\Theta(n^r)$ . So far this conjecture is known to be true only for rationals of form $1+1/k$ and $2-1/k$ , for integers $k\geq 2$ . In this paper, we add a new form of rationals for which the conjecture is true: $2-2/(2k+1)$ , for $k\geq 2$ . This in turn also gives an affirmative answer to a question of Pinchasi and Sharir on cube-like graphs. Recently, a version of Erdős and Simonovits $^{\prime}$ s conjecture, where one replaces a single graph by a finite family, was confirmed by Bukh and Conlon. They proposed a construction of bipartite graphs which should satisfy Erdős and Simonovits $^{\prime}$ s conjecture. Our result can also be viewed as a first step towards verifying Bukh and Conlon $^{\prime}$ s conjecture. We also prove an upper bound on the Turán number of theta graphs in an asymmetric setting and employ this result to obtain another new rational exponent for Turán exponents: $r=7/5$ .

An Entropy Approach to the Hard-Core Model on Bipartite Graphs

2001 ◽  
Vol 10 (3) ◽  
pp. 219-237 ◽  
Author(s):  
JEFF KAHN
Keyword(s):  
Phase Transition ◽  
Bipartite Graph ◽  
Upper Bound ◽  
Hard Core ◽  
Independent Set ◽  
Independent Sets ◽  
Bipartite Graphs ◽  
Core Model ◽  
Hard Core Model ◽  
Regular Bipartite Graph

We use entropy ideas to study hard-core distributions on the independent sets of a finite, regular bipartite graph, specifically distributions according to which each independent set I is chosen with probability proportional to λ[mid ]I[mid ] for some fixed λ > 0. Among the results obtained are rather precise bounds on occupation probabilities; a ‘phase transition’ statement for Hamming cubes; and an exact upper bound on the number of independent sets in an n-regular bipartite graph on a given number of vertices.

scholarly journals Remarks on the upper bound for the Randić energy of bipartite graphs

2017 ◽  
Vol 221 ◽  
pp. 67-70
Author(s):  
Edin Glogić ◽  
Emir Zogić ◽  
Nataša Glišović
Keyword(s):  
Upper Bound ◽  
Bipartite Graphs

scholarly journals A Generalized Information-Theoretic Approach for Bounding the Number of Independent Sets in Bipartite Graphs

Entropy ◽  
2021 ◽  
Vol 23 (3) ◽  
pp. 270
Author(s):  
Igal Sason
Keyword(s):  
Upper Bound ◽  
Independent Sets ◽  
Bipartite Graphs ◽  
Theoretic Approach ◽  
Regular Graphs ◽  
Proof Technique ◽  
Information Theoretic ◽  
Theoretic Proof ◽  
Information Theoretic Approach ◽  
General Graphs

This paper studies the problem of upper bounding the number of independent sets in a graph, expressed in terms of its degree distribution. For bipartite regular graphs, Kahn (2001) established a tight upper bound using an information-theoretic approach, and he also conjectured an upper bound for general graphs. His conjectured bound was recently proved by Sah et al. (2019), using different techniques not involving information theory. The main contribution of this work is the extension of Kahn’s information-theoretic proof technique to handle irregular bipartite graphs. In particular, when the bipartite graph is regular on one side, but may be irregular on the other, the extended entropy-based proof technique yields the same bound as was conjectured by Kahn (2001) and proved by Sah et al. (2019).

scholarly journals A Paintability Version of the Combinatorial Nullstellensatz, and List Colorings of $k$-partite $k$-uniform Hypergraphs

10.37236/448 ◽  
2010 ◽  
Vol 17 (1) ◽  
Author(s):  
Uwe Schauz
Keyword(s):  
Upper Bound ◽  
Bipartite Graphs ◽  
Combinatorial Nullstellensatz ◽  
List Coloring ◽  
Line List ◽  
Uniform Hypergraphs ◽  
Coloring Number ◽  
Subject Specific ◽  
List Colorings ◽  
On Line

We study the list coloring number of $k$-uniform $k$-partite hypergraphs. Answering a question of Ramamurthi and West, we present a new upper bound which generalizes Alon and Tarsi's bound for bipartite graphs, the case $k=2$. Our results hold even for paintability (on" line list colorability). To prove this additional strengthening, we provide a new subject"=specific version of the Combinatorial Nullstellensatz.

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