scholarly journals A note on PM-compact $ K_4 $-free bricks

2021 ◽  
Vol 7 (3) ◽  
pp. 3648-3652
Author(s):  
Jinqiu Zhou ◽  
◽  
Qunfang Li ◽  
Keyword(s):  
Connected Graph ◽  
Perfect Matching ◽  
Building Blocks ◽  
Discrete Mathematics ◽  
Siam Journal ◽  
Multiple Edges ◽  
Generation Procedure

<abstract><p>A 3-connected graph is a <italic>brick</italic> if the graph obtained from it by deleting any two distinct vertices has a perfect matching. The importance of bricks stems from the fact that they are building blocks of the matching covered graphs. Lovász (Combinatorica, 3 (1983), 105-117) showed that every brick is $ K_4 $-based or $ \overline{C}_6 $-based. A brick is <italic>$ K_4 $-free</italic> (respectively, <italic>$ \overline{C}_6 $-free</italic>) if it is not $ K_4 $-based (respectively, $ \overline{C}_6 $-based). Recently, Carvalho, Lucchesi and Murty (SIAM Journal on Discrete Mathematics, 34(3) (2020), 1769-1790) characterised the PM-compact $ \overline{C}_6 $-free bricks. In this note, we show that, by using the brick generation procedure established by Norine and Thomas (J Combin Theory Ser B, 97 (2007), 769-817), the only PM-compact $ K_4 $-free brick is $ \overline{C}_6 $, up to multiple edges.</p></abstract>

K − Bipartite Matching Extendability of Special Graphs

2011 ◽  
Vol 121-126 ◽  
pp. 4008-4012
Author(s):  
Zhi Hao Hui ◽  
Jin Wei Yang ◽  
Biao Zhao
Keyword(s):  
Positive Integer ◽  
Connected Graph ◽  
Perfect Matching ◽  
Bipartite Matching ◽  
Simple Connected Graph ◽  
Matching Extendability

Let be a simple connected graph containing a perfect matching. For a positive integer , , is said to be bipartite matching extendable if every bipartite matching of with is included in a perfect matching of . In this paper, we show that bipartite matching extendability of some special graphs.

scholarly journals Undirecting membership in models of Anti-Foundation

2020 ◽  
Author(s):  
Bea Adam-Day ◽  
Peter J. Cameron
Keyword(s):  
Set Theory ◽  
Random Graph ◽  
Connected Graph ◽  
Countable Model ◽  
Connected Components ◽  
Membership Relation ◽  
Multiple Edges

AbstractIt is known that, if we take a countable model of Zermelo–Fraenkel set theory ZFC and “undirect” the membership relation (that is, make a graph by joining x to y if either $$x\in y$$ x ∈ y or $$y\in x$$ y ∈ x ), we obtain the Erdős–Rényi random graph. The crucial axiom in the proof of this is the Axiom of Foundation; so it is natural to wonder what happens if we delete this axiom, or replace it by an alternative (such as Aczel’s Anti-Foundation Axiom). The resulting graph may fail to be simple; it may have loops (if $$x\in x$$ x ∈ x for some x) or multiple edges (if $$x\in y$$ x ∈ y and $$y\in x$$ y ∈ x for some distinct x, y). We show that, in ZFA, if we keep the loops and ignore the multiple edges, we obtain the “random loopy graph” (which is $$\aleph _0$$ ℵ 0 -categorical and homogeneous), but if we keep multiple edges, the resulting graph is not $$\aleph _0$$ ℵ 0 -categorical, but has infinitely many 1-types. Moreover, if we keep only loops and double edges and discard single edges, the resulting graph contains countably many connected components isomorphic to any given finite connected graph with loops.

scholarly journals Extremal Values of Variable Sum Exdeg Index for Conjugated Bicyclic Graphs

2021 ◽  
Vol 2021 ◽  
pp. 1-11
Author(s):  
Muhammad Rizwan ◽  
Akhlaq Ahmad Bhatti ◽  
Muhammad Javaid ◽  
Ebenezer Bonyah
Keyword(s):  
Connected Graph ◽  
Perfect Matching ◽  
Matching Number ◽  
Bicyclic Graph ◽  
Bicyclic Graphs ◽  
Extremal Values

A connected graph G V , E in which the number of edges is one more than its number of vertices is called a bicyclic graph. A perfect matching of a graph is a matching in which every vertex of the graph is incident to exactly one edge of the matching set such that the number of vertices is two times its matching number. In this paper, we investigated maximum and minimum values of variable sum exdeg index, SEI a for bicyclic graphs with perfect matching for k ≥ 5 and a > 1 .

scholarly journals Lai's Conditions for Spanning and Dominating Closed Trails

10.37236/4511 ◽  
2015 ◽  
Vol 22 (1) ◽  
Author(s):  
Wei-Guo Chen ◽  
Zhi-Hong Chen ◽  
Mei Lu
Keyword(s):  
Graph Theory ◽  
Real Number ◽  
Connected Graph ◽  
Finite Family ◽  
Discrete Mathematics ◽  
Line Graphs ◽  
Connected Graphs ◽  
Supereulerian Graphs ◽  
The Family ◽  
Hamiltonian Line Graphs

A graph is supereulerian if it has a spanning closed trail. For an integer $r$, let ${\cal Q}_0(r)$ be  the family of 3-edge-connected nonsupereulerian graphs of order at most $r$. For a graph $G$, define $\delta_L(G)=\min\{\max\{d(u), d(v) \}| \  \mbox{ for any $uv\in E(G)$} \}$. For a given integer $p\ge 2$ and a given real number $\epsilon$,  a graph $G$ of order $n$ is said to satisfy a Lai's condition if $\delta_L(G)\ge \frac{n}{p}-\epsilon$.  In this paper, we show that  if $G$ is  a  3-edge-connected graph of order $n$ with $\delta_L(G)\ge \frac{n}{p}-\epsilon$, then there is an integer $N(p, \epsilon)$ such that when $n> N(p,\epsilon)$, $G$ is supereulerian if and only if $G$ is not  a graph obtained from a  graph $G_p$ in the finite family ${\cal Q}_0(3p-5)$ by replacing some vertices in $G_p$ with nontrivial graphs. Results on the best possible Lai's  conditions for Hamiltonian line graphs of 3-edge-connected graphs or 3-edge-connected supereulerian graphs are given,  which are improvements of the results in [J. Graph Theory 42(2003) 308-319] and in [Discrete Mathematics, 310(2010) 2455-2459].

scholarly journals On a problem of ore on maximal trees

1970 ◽  
Vol 11 (3) ◽  
pp. 379-380 ◽  
Author(s):  
S. B. Rao
Keyword(s):  
Connected Graph ◽  
Maximal Tree ◽  
Finite Case ◽  
Multiple Edges

We consider only graphs without loops or multiple edges. Pertinent definitions are given below. For notation and other definitions we generally follow Ore [1].A connected graph G = (X, E) is said to have the property P if for every maximal tree T of G there exists a vertex aT of G such that distance between aT and x is same in T as in G for every x in X. The following problem has been posed by Ore (see [1] page 103, problem 4): Determine the graphs with property P. This paper presents a solution to the above problem in the finite case.

scholarly journals 3-Connected Cores In Random Planar Graphs

2011 ◽  
Vol 20 (3) ◽  
pp. 381-412 ◽  
Author(s):  
NIKOLAOS FOUNTOULAKIS ◽  
KONSTANTINOS PANAGIOTOU
Keyword(s):  
Planar Graph ◽  
Planar Graphs ◽  
General Theorem ◽  
Building Blocks ◽  
Discrete Mathematics ◽  
List Type ◽  
Type Number ◽  
Side Constraints ◽  
Evaluating Methods ◽  
Sharp Concentration

The study of the structural properties of large random planar graphs has become in recent years a field of intense research in computer science and discrete mathematics. Nowadays, a random planar graph is an important and challenging model for evaluating methods that are developed to study properties of random graphs from classes with structural side constraints.In this paper we focus on the structure of random 2-connected planar graphs regarding the sizes of their 3-connected building blocks, which we callcores. In fact, we prove a general theorem regarding random biconnected graphs from various classes. IfBnis a graph drawn uniformly at random from a suitable classof labelled biconnected graphs, then we show that with probability 1 −o(1) asn→ ∞,Bnbelongs to exactly one of the following categories:(i)either there is a uniquegiantcore inBn, that is, there is a 0 <c=c() < 1 such that the largest core contains ~cnvertices, and every other core contains at mostnαvertices, where 0 < α = α() < 1;(ii)or all cores ofBncontainO(logn) vertices.Moreover, we find the critical condition that determines the category to whichBnbelongs, and also provide sharp concentration results for the counts of cores of all sizes between 1 andn. As a corollary, we obtain that a random biconnected planar graph belongs to category (i), where in particularc= 0.765. . . and α = 2/3.

scholarly journals On the Harary index of cacti

Filomat ◽  
2014 ◽  
Vol 28 (3) ◽  
pp. 493-507 ◽  
Author(s):  
Zhongxun Zhu ◽  
Ting Tao ◽  
Jing Yu ◽  
Liansheng Tan
Keyword(s):  
Connected Graph ◽  
Perfect Matching ◽  
Upper Bounds ◽  
Common Vertex ◽  
Harary Index

The Harary index is defined as the sum of reciprocals of distances between all pairs of vertices of a connected graph. A connected graph G is a cactus if any two of its cycles have at most one common vertex. Let G(n, r) be the set of cacti of order n and with r cycles, ?(2n,r) the set of cacti of order 2n with a perfect matching and r cycles. In this paper, we give the sharp upper bounds of the Harary index of cacti among G (n,r) and ?(2n, r), respectively, and characterize the corresponding extremal cactus.

scholarly journals Π(G,x) Polynomial and Π(G) Index of Armchair Polyhex Nanotubes TUAC6[m,n]

2014 ◽  
Vol 36 ◽  
pp. 201-206
Author(s):  
Mohammad Reza Farahani
Keyword(s):  
Connected Graph ◽  
Infinite Class ◽  
Vertex Set ◽  
Omega Polynomial ◽  
Multiple Edges ◽  
Simple Connected Graph

Let G be a simple connected graph with the vertex set V = V(G) and the edge set E = E(G), without loops and multiple edges. For counting qoc strips in G, Omega polynomial was introduced by Diudea and was defined as Ω(G,x) = ∑cm(G,c)xc where m(G,c) be number of qoc strips of length c in the graph G. Following Omega polynomial, the Sadhana polynomial was defined by Ashrafi et al as Sd(G,x) = ∑cm(G,c)x[E(G)]-c in this paper we compute the Pi polynomial Π(G,x) = ∑cm(G,c)x[E(G)]-c and Pi Index Π(G ) = ∑cc·m(G,c)([E(G)]-c) of an infinite class of “Armchair polyhex nanotubes TUAC6[m,n]”.

Linear-Time Approximation Algorithms for the Max Cut Problem

1993 ◽  
Vol 2 (2) ◽  
pp. 201-210 ◽  
Author(s):  
Nguyen van Ngoc ◽  
Zsolt Tuza
Keyword(s):  
Lower Bound ◽  
Approximation Algorithms ◽  
Lower Bounds ◽  
Connected Graph ◽  
Linear Time ◽  
Worst Case ◽  
Time Approximation ◽  
Bipartite Subgraph ◽  
Max Cut Problem ◽  
Multiple Edges

Let G be a connected graph with n vertices and m edges (multiple edges allowed), and let k ≥ 2 be an integer. There is an algorithm with (optimal) running time of O(m) that finds(i) a bipartite subgraph of G with ≥ m/2 + (n − 1)/4 edges,(ii) a bipartite subgraph of G with ≥ m/2 + 3(n−1)/8 edges if G is triangle-free,(iii) a k-colourable subgraph of G with ≥ m − m/k + (n−1)/k + (k − 3)/2 edges if k ≥ 3 and G is not k-colorable.Infinite families of graphs show that each of those lower bounds on the worst-case performance are best possible (for every algorithm). Moreover, even if short cycles are excluded, the general lower bound of m − m/k cannot be replaced by m − m/k + εm for any fixed ε > 0; and it is NP-complete to decide whether a graph with m edges contains a k-colorable subgraph with more than m − m/k + εm edges, for any k ≥ 2 and ε> 0, ε < 1/k.

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