scholarly journals An extremal problem for a graphic sequence to have a realization containing every 2-tree with prescribed size

2016 ◽  
Vol Vol. 17 no. 3 (Graph Theory) ◽  
Author(s):  
De-Yan Zeng ◽  
Jian-Hua Yin
Keyword(s):  
Lower Bound ◽  
Extremal Problem ◽  
Simple Graph ◽  
Graphic Sequence ◽  
International Audience

International audience A graph $G$ is a $2$<i>-tree</i> if $G=K_3$, or $G$ has a vertex $v$ of degree 2, whose neighbors are adjacent, and $G-v$ is a 2-tree. Clearly, if $G$ is a 2-tree on $n$ vertices, then $|E(G)|=2n-3$. A non-increasing sequence $\pi =(d_1, \ldots ,d_n)$ of nonnegative integers is a <i>graphic sequence</i> if it is realizable by a simple graph $G$ on $n$ vertices. Yin and Li (Acta Mathematica Sinica, English Series, 25(2009)795&#x2013;802) proved that if $k \geq 2$, $n \geq \frac{9}{2}k^2 + \frac{19}{2}k$ and $\pi =(d_1, \ldots ,d_n)$ is a graphic sequence with $\sum \limits_{i=1}^n d_i > (k-2)n$, then $\pi$ has a realization containing every tree on $k$ vertices as a subgraph. Moreover, the lower bound $(k-2)n$ is the best possible. This is a variation of a conjecture due to Erd&#x0151;s and S&oacute;s. In this paper, we investigate an analogue extremal problem for 2-trees and prove that if $k \geq 3$, $n \geq 2k^2-k$ and $\pi =(d_1, \ldots ,d_n)$ is a graphic sequence with $\sum \limits_{i=1}^n d_i > \frac{4kn}{3} - \frac{5n}{3}$ then $\pi$ has a realization containing every 2-tree on $k$ vertices as a subgraph. We also show that the lower bound $\frac{4kn}{3} - \frac{5n}{3}$ is almost the best possible.

scholarly journals A further result on the potential-Ramsey number of G1 and G2

Filomat ◽  
2019 ◽  
Vol 33 (6) ◽  
pp. 1605-1617
Author(s):  
Jinzhi Du ◽  
Jianhua Yin
Keyword(s):  
Lower Bound ◽  
Complete Graph ◽  
Degree Sequence ◽  
Simple Graph ◽  
Negative Integer ◽  
Ramsey Number ◽  
Complementary Sequence ◽  
Split Graph ◽  
Graphic Sequence

A non-increasing sequence ? = (d1,. . ., dn) of nonnegative integers is a graphic sequence if it is realizable by a simple graph G on n vertices. In this case, G is referred to as a realization of ?. Given a graph H, a graphic sequence ? is potentially H-graphic if ? has a realization containing H as a subgraph. Busch et al. (Graphs Combin., 30(2014)847-859) considered a degree sequence analogue to classical graph Ramsey number as follows: for graphs G1 and G2, the potential-Ramsey number rpot(G1,G2) is the smallest non-negative integer k such that for any k-term graphic sequence ?, either ? is potentially G1-graphic or the complementary sequence ? = (k - 1 - dk,..., k - 1 - d1) is potentially G2-graphic. They also gave a lower bound on rpot(G;Kr+1) for a number of choices of G and determined the exact values for rpot(Kn;Kr+1), rpot(Cn;Kr+1) and rpot(Pn,Kr+1). In this paper, we will extend the complete graph Kr+1 to the complete split graph Sr,s = Kr ? Ks. Clearly, Sr,1 = Kr+1. We first give a lower bound on rpot(G, Sr,s) for a number of choices of G, and then determine the exact values for rpot(Cn, Sr,s) and rpot(Pn, Sr,s).

scholarly journals On Greedy Trie Execution

2012 ◽  
Vol DMTCS Proceedings vol. AQ,... (Proceedings) ◽  
Author(s):  
Zbigniew Gołębiewski ◽  
Filip Zagórski
Keyword(s):  
Lower Bound ◽  
Greedy Algorithm ◽  
International Audience ◽  
The Greedy Algorithm

International audience In the paper "How to select a looser'' Prodinger was analyzing an algorithm where $n$ participants are selecting a leader by flipping <underline>fair</underline> coins, where recursively, the 0-party (those who i.e. have tossed heads) continues until the leader is chosen. We give an answer to the question stated in the Prodinger's paper – what happens if not a 0-party is recursively looking for a leader but always a party with a smaller cardinality. We show the lower bound on the number of rounds of the greedy algorithm (for <underline>fair</underline> coin).

scholarly journals On Total Vertex Irregularity Strength of Hexagonal Cluster Graphs

2021 ◽  
Vol 2021 ◽  
pp. 1-9
Author(s):  
Nurdin Hinding ◽  
Hye Kyung Kim ◽  
Nurtiti Sunusi ◽  
Riskawati Mise
Keyword(s):  
Lower Bound ◽  
Positive Integer ◽  
Simple Graph ◽  
Vertex Set ◽  
Cluster Graph ◽  
Cluster Graphs ◽  
Irregularity Strength

For a simple graph G with a vertex set V G and an edge set E G , a labeling f : V G ∪ ​ E G ⟶ 1,2 , ⋯ , k is called a vertex irregular total k − labeling of G if for any two different vertices x and y in V G we have w t x ≠ w t y where w t x = f x + ∑ u ∈ V G f x u . The smallest positive integer k such that G has a vertex irregular total k − labeling is called the total vertex irregularity strength of G , denoted by tvs G . The lower bound of tvs G for any graph G have been found by Baca et. al. In this paper, we determined the exact value of the total vertex irregularity strength of the hexagonal cluster graph on n cluster for n ≥ 2 . Moreover, we show that the total vertex irregularity strength of the hexagonal cluster graph on n cluster is 3 n 2 + 1 / 2 .

scholarly journals On Vertices of Degree n in Minimally n-Edge-Connected Graphs

1995 ◽  
Vol 4 (1) ◽  
pp. 81-95 ◽  
Author(s):  
W. Mader
Keyword(s):  
Lower Bound ◽  
Simple Graph ◽  
Simple Graphs ◽  
Connected Graphs ◽  
Vertex Number

Let G be a minimally n-edge-connected finite simple graph with vertex number |G| ≥ 2n + 2 + [3/n] and let n ≥ 3 be odd. It is proved that the number of vertices of degree n in G is at least ((n − 1 − ∈n)/(2n + 1))|G| + 2 + 2∈n, where ∈n = (3n + 3)/(2n2 − 3n − 3), and that for every n ≡ 3 (mod 4) this lower bound is attained by infinitely many minimally n-edge-connected finite simple graphs.

scholarly journals A note on domino treewidth

1999 ◽  
Vol Vol. 3 no. 4 ◽  
Author(s):  
Hans L. Bodlaender
Keyword(s):  
Lower Bound ◽  
Simple Proof ◽  
Maximum Degree ◽  
International Audience

International audience In [DO95], Ding and Oporowski proved that for every k, and d, there exists a constant c_k,d, such that every graph with treewidth at most k and maximum degree at most d has domino treewidth at most c_k,d. This note gives a new simple proof of this fact, with a better bound for c_k,d, namely (9k+7)d(d+1) -1. It is also shown that a lower bound of Ω (kd) holds: there are graphs with domino treewidth at least 1/12 × kd-1, treewidth at most k, and maximum degree at most d, for many values k and d. The domino treewidth of a tree is at most its maximum degree.

scholarly journals The Hodge Structure of the Coloring Complex of a Hypergraph (Extended Abstract)

2010 ◽  
Vol DMTCS Proceedings vol. AN,... (Proceedings) ◽  
Author(s):  
Sarah C Rundell ◽  
Jane H Long
Keyword(s):  
Euler Characteristic ◽  
Homology Group ◽  
Hodge Structure ◽  
Hodge Decomposition ◽  
Simple Graph ◽  
Absolute Value ◽  
International Audience ◽  
The Absolute ◽  
Nous Examinons ◽  
Coloring Complex

International audience Let $G$ be a simple graph with $n$ vertices. The coloring complex$ Δ (G)$ was defined by Steingrímsson, and the homology of $Δ (G)$ was shown to be nonzero only in dimension $n-3$ by Jonsson. Hanlon recently showed that the Eulerian idempotents provide a decomposition of the homology group $H_{n-3}(Δ (G))$ where the dimension of the $j^th$ component in the decomposition, $H_{n-3}^{(j)}(Δ (G))$, equals the absolute value of the coefficient of $λ ^j$ in the chromatic polynomial of $G, _{\mathcal{χg}}(λ )$. Let $H$ be a hypergraph with $n$ vertices. In this paper, we define the coloring complex of a hypergraph, $Δ (H)$, and show that the coefficient of $λ ^j$ in $χ _H(λ )$ gives the Euler Characteristic of the $j^{th}$ Hodge subcomplex of the Hodge decomposition of $Δ (H)$. We also examine conditions on a hypergraph, $H$, for which its Hodge subcomplexes are Cohen-Macaulay, and thus where the absolute value of the coefficient of $λ ^j$ in $χ _H(λ )$ equals the dimension of the $j^{th}$ Hodge piece of the Hodge decomposition of $Δ (H)$. Soit $G$ un graphe simple à n sommets. Le complexe de coloriage $Δ (G)$ a été défini par Steingrímsson et Jonsson a prouvé que l'homologie de $Δ (G)$ est non nulle seulement en dimension $n-3$. Hanlon a récemment prouvé que les idempotents eulériens fournissent une décomposition du groupe d'homologie $H_{n-3}(Δ (G))$ où la dimension de la $j^e$ composante dans la décomposition de $H_{n-3}^{(j)}(Δ (G))$ est égale à la valeur absolue du coefficient de $λ ^j$ dans le polynôme chromatique de $G, _{\mathcal{χg}}(λ )$ . Soit H un hypergraphe à $n$ sommets. Dans ce texte, nous définissons le complexe de coloration d'un hypergraphe $Δ (H)$ et nous prouvons que le coefficient de $λ ^j$ dans $χ _H(λ )$ donne la caractéristique d'Euler du $j^e$ sous-complexe de Hodge dans la décomposition de Hodge de Δ (H). Nous examinons également des conditions sur un hypergraphe H pour lesquelles les sous-complexes de Hodge sont Cohen-Macaulay. Ainsi la valeur absolue du coefficient de $λ ^j$ in $χ _H(λ )$ est égale à la dimension du $j^e$sous-complexe de Hodge dans la décomposition de Hodge de $Δ (H)$.

scholarly journals On the sum-connectivity index

Filomat ◽  
2011 ◽  
Vol 25 (3) ◽  
pp. 29-42 ◽  
Author(s):  
Shilin Wang ◽  
Zhou Bo ◽  
Nenad Trinajstic
Keyword(s):  
Lower Bound ◽  
Minimum Degree ◽  
Simple Graph ◽  
Connectivity Index ◽  
Extremal Graphs ◽  
Free Graph ◽  
Mathematical Chemistry ◽  
Degree Of Vertex

The sum-connectivity index of a simple graph G is defined in mathematical chemistry as R+(G) = ? uv?E(G)(du+dv)?1/2, where E(G) is the edge set of G and du is the degree of vertex u in G. We give a best possible lower bound for the sum-connectivity index of a graph (a triangle-free graph, respectively) with n vertices and minimum degree at least two and characterize the extremal graphs, where n ? 11.

scholarly journals The short toric polynomial

2011 ◽  
Vol DMTCS Proceedings vol. AO,... (Proceedings) ◽  
Author(s):  
Gábor Hetyei
Keyword(s):  
Lower Bound ◽  
Reflection Principle ◽  
Lattice Path ◽  
Simple Polytope ◽  
Independent Form ◽  
International Audience ◽  
Bound Theorem ◽  
Lattice Path Enumeration ◽  
Nous Utilisons ◽  
Eulerian Poset

International audience We introduce the short toric polynomial associated to a graded Eulerian poset. This polynomial contains the same information as Stanley's pair of toric polynomials, but allows different algebraic manipulations. Stanley's intertwined recurrence may be replaced by a single recurrence, in which the degree of the discarded terms is independent of the rank. A short toric variant of the formula by Bayer and Ehrenborg, expressing the toric h-vector in terms of the cd-index, may be stated in a rank-independent form, and it may be shown using weighted lattice path enumeration and the reflection principle. We use our techniques to derive a formula expressing the toric h-vector of a dual simplicial Eulerian poset in terms of its f-vector. This formula implies Gessel's formula for the toric h-vector of a cube, and may be used to prove that the nonnegativity of the toric h-vector of a simple polytope is a consequence of the Generalized Lower Bound Theorem holding for simplicial polytopes. Nous introduisons le polynôme torique court associé à un ensemble ordonné Eulérien. Ce polynôme contient la même information que le couple de polynômes toriques de Stanley, mais il permet des manipulations algébriques différentes. La récurrence entrecroisée de Stanley peut être remplacée par une seule récurrence dans laquelle le degré des termes écartés est indépendant du rang. La variante torique courte de la formule de Bayer et Ehrenborg, qui exprime le vecteur torique d'un ensemble ordonné Eulérien en termes de son cd-index, est énoncée sous une forme qui ne dépend pas du rang et qui peut être démontrée en utilisant une énumération des chemins pondérés et le principe de réflexion. Nous utilisons nos techniques pour dériver une formule exprimant le vecteur h-torique d'un ensemble ordonné Eulérien dont le dual est simplicial, en termes de son f-vecteur. Cette formule implique la formule de Gessel pour le vecteur h-torique d'un cube, et elle peut être utilisée pour démontrer que la positivité du vecteur h-torique d'un polytope simple est une conséquence du Théorème de la Borne Inférieure Généralisé appliqué aux polytopes simpliciaux.

scholarly journals Packing Plane Perfect Matchings into a Point Set

2015 ◽  
Vol Vol. 17 no.2 (Graph Theory) ◽  
Author(s):  
Ahmad Biniaz ◽  
Prosenjit Bose ◽  
Anil Maheshwari ◽  
Michiel Smid
Keyword(s):  
Lower Bound ◽  
General Position ◽  
Perfect Matchings ◽  
Point Sets ◽  
Exact Answer ◽  
International Audience ◽  
Point Set

International audience Given a set $P$ of $n$ points in the plane, where $n$ is even, we consider the following question: How many plane perfect matchings can be packed into $P$? For points in general position we prove the lower bound of &#x230A;log<sub>2</sub>$n$&#x230B;$-1$. For some special configurations of point sets, we give the exact answer. We also consider some restricted variants of this problem.

scholarly journals Edge-Removal and Non-Crossing Configurations in Geometric Graphs

2010 ◽  
Vol Vol. 12 no. 1 (Graph and Algorithms) ◽  
Author(s):  
Oswin Aichholzer ◽  
Sergio Cabello ◽  
Ruy Fabila-Monroy ◽  
David Flores-Peñaloza ◽  
Thomas Hackl ◽  
...  
Keyword(s):  
Extremal Problem ◽  
General Position ◽  
Geometric Graph ◽  
Perfect Matchings ◽  
Geometric Graphs ◽  
Line Segments ◽  
Straight Line ◽  
Edge Removal ◽  
International Audience ◽  
Point Set

Graphs and Algorithms International audience A geometric graph is a graph G = (V, E) drawn in the plane, such that V is a point set in general position and E is a set of straight-line segments whose endpoints belong to V. We study the following extremal problem for geometric graphs: How many arbitrary edges can be removed from a complete geometric graph with n vertices such that the remaining graph still contains a certain non-crossing subgraph. The non-crossing subgraphs that we consider are perfect matchings, subtrees of a given size, and triangulations. In each case, we obtain tight bounds on the maximum number of removable edges.

Sign in / Sign up

Export Citation Format

Share Document