scholarly journals G-Intersection Theorems for Matchings and Other Graphs

2008 ◽  
Vol 17 (4) ◽  
pp. 559-575
Author(s):  
J. ROBERT JOHNSON ◽  
JOHN TALBOT
Keyword(s):  
Phase Transition ◽  
Positive Root ◽  
Sufficient Condition ◽  
Matching Number ◽  
Sharp Transition ◽  
Intersecting Family ◽  
Vertex Set ◽  
Almost All ◽  
Sharp Phase Transition

If G is a graph with vertex set [n] then $\mathcal{A}\subseteq 2^{[n]}$ is G-intersecting if, for all $A,B\in \mathcal{A}$, either A ∩ B ≠ ∅ or there exist a ∈ A and b ∈ B such that a ~Gb.The question of how large a k-uniform G-intersecting family can be was first considered by Bohman, Frieze, Ruszinkó and Thoma [2], who identified two natural candidates for the extrema depending on the relative sizes of k and n and asked whether there is a sharp phase transition between the two. Our first result shows that there is a sharp transition and characterizes the extremal families when G is a matching. We also give an example demonstrating that other extremal families can occur.Our second result gives a sufficient condition for the largest G-intersecting family to contain almost all k-sets. In particular we show that if Cn is the n-cycle and k > αn + o(n), where α = 0.266. . . is the smallest positive root of (1 − x)3(1 + x) = 1/2, then the largest Cn-intersecting family has size $(1-o(1))\binom{n}{k}$.Finally we consider the non-uniform problem, and show that in this case the size of the largest G-intersecting family depends on the matching number of G.

scholarly journals The Maximum Number of Cliques in Hypergraphs without Large Matchings

10.37236/9604 ◽  
2020 ◽  
Vol 27 (4) ◽  
Author(s):  
Erica L.L. Liu ◽  
Jian Wang
Keyword(s):  
Sufficient Condition ◽  
Matching Number ◽  
Uniform Hypergraph ◽  
Rainbow Matching ◽  
Uniform Hypergraphs ◽  
Large N ◽  
Vertex Set

Let $[n]$ denote the set $\{1, 2, \ldots, n\}$ and $\mathcal{F}^{(r)}_{n,k,a}$ be an $r$-uniform hypergraph on the vertex set $[n]$ with edge set consisting of all the $r$-element subsets of $[n]$ that contains at least $a$ vertices in $[ak+a-1]$. For $n\geq 2rk$, Frankl proved that $\mathcal{F}^{(r)}_{n,k,1}$ maximizes the number of edges in $r$-uniform hypergraphs on $n$ vertices with the matching number at most $k$. Huang, Loh and Sudakov considered a multicolored version of the Erd\H{o}s matching conjecture, and provided a sufficient condition on the number of edges for a multicolored hypergraph to contain a rainbow matching of size $k$. In this paper, we show that $\mathcal{F}^{(r)}_{n,k,a}$ maximizes the number of $s$-cliques in $r$-uniform hypergraphs on $n$ vertices with the matching number at most $k$ for sufficiently large $n$, where $a=\lfloor \frac{s-r}{k} \rfloor+1$. We also obtain a condition on the number of $s$-clques for a multicolored $r$-uniform hypergraph to contain a rainbow matching of size $k$, which reduces to the condition of Huang, Loh and Sudakov when $s=r$.

scholarly journals Cycle partitions of regular graphs

2020 ◽  
pp. 1-24
Author(s):  
Vytautas Gruslys ◽  
Shoham Letzter
Keyword(s):  
Regular Graph ◽  
Bipartite Graphs ◽  
Disjoint Paths ◽  
Regular Graphs ◽  
Simple Construction ◽  
Vertex Set ◽  
Almost All ◽  
Vertex Disjoint

Abstract Magnant and Martin conjectured that the vertex set of any d-regular graph G on n vertices can be partitioned into $n / (d+1)$ paths (there exists a simple construction showing that this bound would be best possible). We prove this conjecture when $d = \Omega(n)$ , improving a result of Han, who showed that in this range almost all vertices of G can be covered by $n / (d+1) + 1$ vertex-disjoint paths. In fact our proof gives a partition of V(G) into cycles. We also show that, if $d = \Omega(n)$ and G is bipartite, then V(G) can be partitioned into n/(2d) paths (this bound is tight for bipartite graphs).

scholarly journals Bipartite graphs with close domination and k-domination numbers

2020 ◽  
Vol 18 (1) ◽  
pp. 873-885
Author(s):  
Gülnaz Boruzanlı Ekinci ◽  
Csilla Bujtás
Keyword(s):  
Positive Integer ◽  
Dominating Set ◽  
Domination Number ◽  
Bipartite Graphs ◽  
Sufficient Condition ◽  
Necessary And Sufficient Condition ◽  
Minimum Cardinality ◽  
Vertex Set ◽  
Necessary And Sufficient ◽  
The Given

Abstract Let k be a positive integer and let G be a graph with vertex set V(G) . A subset D\subseteq V(G) is a k -dominating set if every vertex outside D is adjacent to at least k vertices in D . The k -domination number {\gamma }_{k}(G) is the minimum cardinality of a k -dominating set in G . For any graph G , we know that {\gamma }_{k}(G)\ge \gamma (G)+k-2 where \text{Δ}(G)\ge k\ge 2 and this bound is sharp for every k\ge 2 . In this paper, we characterize bipartite graphs satisfying the equality for k\ge 3 and present a necessary and sufficient condition for a bipartite graph to satisfy the equality hereditarily when k=3 . We also prove that the problem of deciding whether a graph satisfies the given equality is NP-hard in general.

scholarly journals On k-rainbow domination in middle graphs

2021 ◽  
Author(s):  
Kijung Kim
Keyword(s):  
Minimum Weight ◽  
Domination Number ◽  
Simple Graph ◽  
Upper And Lower Bounds ◽  
Matching Number ◽  
Domatic Number ◽  
Rainbow Domination ◽  
Vertex Set ◽  
Middle Graph ◽  
Dominating Function

Let $G$ be a finite simple graph with vertex set $V(G)$ and edge set $E(G)$. A function $f : V(G) \rightarrow \mathcal{P}(\{1, 2, \dotsc, k\})$ is a \textit{$k$-rainbow dominating function} on $G$ if for each vertex $v \in V(G)$ for which $f(v)= \emptyset$, it holds that $\bigcup_{u \in N(v)}f(u) = \{1, 2, \dotsc, k\}$. The weight of a $k$-rainbow dominating function is the value $\sum_{v \in V(G)}|f(v)|$. The \textit{$k$-rainbow domination number} $\gamma_{rk}(G)$ is the minimum weight of a $k$-rainbow dominating function on $G$. In this paper, we initiate the study of $k$-rainbow domination numbers in middle graphs. We define the concept of a middle $k$-rainbow dominating function, obtain some bounds related to it and determine the middle $3$-rainbow domination number of some classes of graphs. We also provide upper and lower bounds for the middle $3$-rainbow domination number of trees in terms of the matching number. In addition, we determine the $3$-rainbow domatic number for the middle graph of paths and cycles.

On Spanning and Dominating Circuits in Graphs

1977 ◽  
Vol 20 (2) ◽  
pp. 215-220 ◽  
Author(s):  
L. Lesniak-Foster ◽  
James E. Williamson
Keyword(s):  
Line Graph ◽  
Dominating Set ◽  
Line Graphs ◽  
Sufficient Condition ◽  
Vertex Set ◽  
On Line ◽  
One To One

AbstractA set E of edges of a graph G is said to be a dominating set of edges if every edge of G either belongs to E or is adjacent to an edge of E. If the subgraph 〈E〉 induced by E is a trail T, then T is called a dominating trail of G. Dominating circuits are defined analogously. A sufficient condition is given for a graph to possess a spanning (and thus dominating) circuit and a sufficient condition is given for a graph to possess a spanning (and thus dominating) trail between each pair of distinct vertices. The line graph L(G) of a graph G is defined to be that graph whose vertex set can be put in one-to-one correspondence with the edge set of G in such a way that two vertices of L(G) are adjacent if and only if the corresponding edges of G are adjacent. The existence of dominating trails and circuits is employed to present results on line graphs and second iterated line graphs, respectively.

A study of enhanced power graphs of finite groups

2019 ◽  
Vol 19 (04) ◽  
pp. 2050062 ◽  
Author(s):  
Samir Zahirović ◽  
Ivica Bošnjak ◽  
Rozália Madarász
Keyword(s):  
Finite Groups ◽  
Prime Power ◽  
Cyclic Subgroup ◽  
Nilpotent Groups ◽  
Sufficient Condition ◽  
Finite Abelian Groups ◽  
Power Graph ◽  
Vertex Set ◽  
A Finite Group

The enhanced power graph [Formula: see text] of a group [Formula: see text] is the graph with vertex set [Formula: see text] such that two vertices [Formula: see text] and [Formula: see text] are adjacent if they are contained in the same cyclic subgroup. We prove that finite groups with isomorphic enhanced power graphs have isomorphic directed power graphs. We show that any isomorphism between undirected power graph of finite groups is an isomorphism between enhanced power graphs of these groups, and we find all finite groups [Formula: see text] for which [Formula: see text] is abelian, all finite groups [Formula: see text] with [Formula: see text] being prime power, and all finite groups [Formula: see text] with [Formula: see text] being square-free. Also, we describe enhanced power graphs of finite abelian groups. Finally, we give a characterization of finite nilpotent groups whose enhanced power graphs are perfect, and we present a sufficient condition for a finite group to have weakly perfect enhanced power graph.

Generalized unit and unitary Cayley graphs of finite rings

2019 ◽  
Vol 18 (01) ◽  
pp. 1950006 ◽  
Author(s):  
T. Tamizh Chelvam ◽  
S. Anukumar Kathirvel
Keyword(s):  
Commutative Ring ◽  
Undirected Graph ◽  
Unit Element ◽  
Sufficient Condition ◽  
Necessary And Sufficient Condition ◽  
Finite Rings ◽  
Vertex Set ◽  
Finite Commutative Ring ◽  
Necessary And Sufficient ◽  
Nonzero Identity

Let [Formula: see text] be a finite commutative ring with nonzero identity and [Formula: see text] be the set of all units of [Formula: see text] The graph [Formula: see text] is the simple undirected graph with vertex set [Formula: see text] in which two distinct vertices [Formula: see text] and [Formula: see text] are adjacent if and only if there exists a unit element [Formula: see text] in [Formula: see text] such that [Formula: see text] is a unit in [Formula: see text] In this paper, we obtain degree of all vertices in [Formula: see text] and in turn provide a necessary and sufficient condition for [Formula: see text] to be Eulerian. Also, we give a necessary and sufficient condition for the complement [Formula: see text] to be Eulerian, Hamiltonian and planar.

scholarly journals Concentration-driven phase transition and self-assembly in drying droplets of diluting whole blood

2020 ◽  
Vol 10 (1) ◽  
Author(s):  
Anusuya Pal ◽  
Amalesh Gope ◽  
John D. Obayemi ◽  
Germano S. Iannacchione
Keyword(s):  
Phase Transition ◽  
Self Assembly ◽  
Whole Blood ◽  
Morphological Analysis ◽  
Hierarchical Structures ◽  
Contact Angle Measurement ◽  
Angle Measurement ◽  
Colloidal Systems ◽  
Functional Complexity ◽  
Sharp Phase Transition

Abstract Multi-colloidal systems exhibit a variety of structural and functional complexity owing to their ability to interact amongst different components into self-assembled structures. This paper presents experimental confirmations that reveal an interesting sharp phase transition during the drying state and in the dried film as a function of diluting concentrations ranging from 100% (undiluted whole blood) to 12.5% (diluted concentrations). An additional complementary contact angle measurement exhibits a monotonic decrease with a peak as a function of drying. This peak is related to a change in visco-elasticity that decreases with dilution, and disappears at the dilution concentration for the observed phase transition equivalent to 62% (v/v). This unique behavior is clearly commensurate with the optical image statistics and morphological analysis; and it is driven by the decrease in the interactions between various components within this bio-colloid. The implications of these phenomenal systems may address many open-ended questions of complex hierarchical structures.

Notizen: Fluorine Diffusion and Phase Transition in Superionic Conductor KSn2F5 as Studied by 19F NMR, Electrical Conductivity and DSC

1983 ◽  
Vol 38 (5) ◽  
pp. 593-594 ◽  
Author(s):  
W. D. Basler ◽  
I. V. Murin ◽  
S. V. Chernov
Keyword(s):  
Magnetic Field ◽  
Phase Transition ◽  
Activation Energy ◽  
Electrical Conductivity ◽  
Relaxation Time ◽  
Field Gradient ◽  
Magnetic Field Gradient ◽  
Pulsed Magnetic Field ◽  
19F Nmr ◽  
Sharp Transition

The diffusion of fluorine in KSn2F5 has been studied by T1 and T2 relaxation time measurements of 19F NMR (200-500 K) and pulsed magnetic Field gradient tech­niques (390-480 K). Near 423 K a sharp transition into the superionic state has been found, the fluorine diffusion increasing by a factor of 4 within a range of 3 K. Conduc­tivity measurements only show a change in the activation energy.

CHARACTERISATION OF Al DEFECTS IN SmBa2Cu3-XAlXO6+δ SUPERCONDUCTOR

2003 ◽  
Vol 17 (04n06) ◽  
pp. 936-941 ◽  
Author(s):  
M. SCAVINI ◽  
L. MOLLICA ◽  
R. BIANCHI ◽  
G. A. COSTA ◽  
M. FERRETTI ◽  
...  
Keyword(s):  
Phase Transition ◽  
Short Range ◽  
Microprobe Analysis ◽  
Electron Microprobe Analysis ◽  
Orthorhombic Phase ◽  
Superconducting Phase ◽  
Al Doping ◽  
X Ray ◽  
Reducing Conditions ◽  
Almost All

We present here a study on the effect of Al doping on the stucture of SmBa 2 Cu 3-X Al X O 6+δ (Sm-123) superconductor. Electron MicroProbe Analysis (EMPA) and X-Ray Powder Diffraction (XRPD) have revealed that the limit of a aluminium solubility x is between 0.5 and 0.6. For further doping BaAl 2 O 4 appears besides the superconducting phase. XRPD analysis on samples annealed in both oxidising and reducing conditions have revealed that the Al doping inhibits the tetragonal to orthorhombic phase transition. Nuclear Magnetic Resonance (NMR) analysis has shown that almost all the Al ions are coordinated tetrahedrally. The comparison between oxygen non-stoichiometry in pure and Al doped SmBa 2 Cu 3-X Al X O 6+δ suggests that the Al ions are ordered in clusters. A model is proposed for short-range order around Al doping ions which allows us to interpret the phase transition inhibition.

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