scholarly journals Constructive Upper Bounds for Cycle-Saturated Graphs of Minimum Size

10.37236/1055 ◽  
2006 ◽  
Vol 13 (1) ◽  
Author(s):  
Ronald Gould ◽  
Tomasz Łuczak ◽  
John Schmitt
Keyword(s):  
Upper Bounds ◽  
Minimum Size ◽  
Saturated Graphs ◽  
Minimum Number

A graph $G$ is said to be $C_l$-saturated if $G$ contains no cycle of length $l$, but for any edge in the complement of $G$ the graph $G+e$ does contain a cycle of length $l$. The minimum number of edges of a $C_l$-saturated graph was shown by Barefoot et al. to be between $n+c_1{n\over l}$ and $n+c_2{n\over l}$ for some positive constants $c_1$ and $c_2$. This confirmed a conjecture of Bollobás. Here we improve the value of $c_2$ for $l \geq 8$.

scholarly journals Saturated Graphs of Prescribed Minimum Degree

2016 ◽  
Vol 26 (2) ◽  
pp. 201-207 ◽  
Author(s):  
A. NICHOLAS DAY
Keyword(s):  
Minimum Degree ◽  
Upper Bounds ◽  
Saturated Graphs ◽  
Minimum Number

A graph G is H-saturated if it contains no copy of H as a subgraph but the addition of any new edge to G creates a copy of H. In this paper we are interested in the function satt(n,p), defined to be the minimum number of edges that a Kp-saturated graph on n vertices can have if it has minimum degree at least t. We prove that satt(n,p) = tn − O(1), where the limit is taken as n tends to infinity. This confirms a conjecture of Bollobás when p = 3. We also present constructions for graphs that give new upper bounds for satt(n,p).

TOTAL OUTER-CONNECTED DOMINATION SUBDIVISION NUMBERS IN GRAPHS

2013 ◽  
Vol 05 (03) ◽  
pp. 1350009
Author(s):  
O. FAVARON ◽  
R. KHOEILAR ◽  
S. M. SHEIKHOLESLAMI
Keyword(s):  
Connected Graph ◽  
Dominating Set ◽  
Domination Number ◽  
Upper Bounds ◽  
Connected Dominating Set ◽  
Minimum Size ◽  
Connected Domination Number ◽  
Minimum Number ◽  
Domination Subdivision Number

A set S of vertices of a graph G is a total outer-connected dominating set if every vertex in V(G) is adjacent to some vertex in S and the subgraph G[V\S] induced by V\S is connected. The total outer-connected domination numberγ toc (G) is the minimum size of such a set. The total outer-connected domination subdivision number sd γ toc (G) is the minimum number of edges that must be subdivided in order to increase the total outer-connected domination number. We prove the existence of sd γ toc (G) for every connected graph G of order at least 3 and give upper bounds on it in some classes of graphs.

Conflict-Free Colourings of Uniform Hypergraphs With Few Edges

2012 ◽  
Vol 21 (4) ◽  
pp. 611-622 ◽  
Author(s):  
A. KOSTOCHKA ◽  
M. KUMBHAT ◽  
T. ŁUCZAK
Keyword(s):  
Chromatic Number ◽  
Upper Bound ◽  
Upper Bounds ◽  
Uniform Hypergraph ◽  
Lower And Upper Bounds ◽  
Uniform Hypergraphs ◽  
Minimum Number

A colouring of the vertices of a hypergraph is called conflict-free if each edge e of contains a vertex whose colour does not repeat in e. The smallest number of colours required for such a colouring is called the conflict-free chromatic number of , and is denoted by χCF(). Pach and Tardos proved that for an (2r − 1)-uniform hypergraph with m edges, χCF() is at most of the order of rm1/r log m, for fixed r and large m. They also raised the question whether a similar upper bound holds for r-uniform hypergraphs. In this paper we show that this is not necessarily the case. Furthermore, we provide lower and upper bounds on the minimum number of edges of an r-uniform simple hypergraph that is not conflict-free k-colourable.

scholarly journals Dicycle Cover of Hamiltonian Oriented Graphs

2016 ◽  
Vol 2016 ◽  
pp. 1-5
Author(s):  
Khalid A. Alsatami ◽  
Hong-Jian Lai ◽  
Xindong Zhang
Keyword(s):  
Bipartite Graphs ◽  
Upper Bounds ◽  
Oriented Graphs ◽  
Minimum Number ◽  
Complete Bipartite Graphs ◽  
Number Of Classes

A dicycle cover of a digraph D is a family F of dicycles of D such that each arc of D lies in at least one dicycle in F. We investigate the problem of determining the upper bounds for the minimum number of dicycles which cover all arcs in a strong digraph. Best possible upper bounds of dicycle covers are obtained in a number of classes of digraphs including strong tournaments, Hamiltonian oriented graphs, Hamiltonian oriented complete bipartite graphs, and families of possibly non-Hamiltonian digraphs obtained from these digraphs via a sequence of 2-sum operations.

scholarly journals On restricted domination in graphs

2007 ◽  
Vol 57 (5) ◽  
Author(s):  
Vladimir Samodivkin
Keyword(s):  
Dominating Set ◽  
Domination Number ◽  
Upper Bounds ◽  
Minimum Number ◽  
Domination In Graphs ◽  
Restricted Domination Number

AbstractThe k-restricted domination number of a graph G is the minimum number d k such that for any subset U of k vertices of G, there is a dominating set in G including U and having at most d k vertices. Some new upper bounds in terms of order and degrees for this number are found.

IDENTIFYING CO-REGULATING MICRORNA GROUPS

2010 ◽  
Vol 08 (01) ◽  
pp. 99-115 ◽  
Author(s):  
JIYUAN AN ◽  
KWOK PUI CHOI ◽  
CHRISTINE A. WELLS ◽  
YI-PING PHOEBE CHEN
Keyword(s):  
Neurodegenerative Diseases ◽  
Target Genes ◽  
Signal To Noise Ratio ◽  
Statistical Tests ◽  
False Positive Rate ◽  
Target Prediction ◽  
Minimum Size ◽  
Supplementary Information ◽  
P Value ◽  
Minimum Number

Background: Current miRNA target prediction tools have the common problem that their false positive rate is high. This renders identification of co-regulating groups of miRNAs and target genes unreliable. In this study, we describe a procedure to identify highly probable co-regulating miRNAs and the corresponding co-regulated gene groups. Our procedure involves a sequence of statistical tests: (1) identify genes that are highly probable miRNA targets; (2) determine for each such gene, the minimum number of miRNAs that co-regulate it with high probability; (3) find, for each such gene, the combination of the determined minimum size of miRNAs that co-regulate it with the lowest p-value; and (4) discover for each such combination of miRNAs, the group of genes that are co-regulated by these miRNAs with the lowest p-value computed based on GO term annotations of the genes. Results: Our method identifies 4, 3 and 2-term miRNA groups that co-regulate gene groups of size at least 3 in human. Our result suggests some interesting hypothesis on the functional role of several miRNAs through a "guilt by association" reasoning. For example, miR-130, miR-19 and miR-101 are known neurodegenerative diseases associated miRNAs. Our 3-term miRNA table shows that miR-130/19/101 form a co-regulating group of rank 22 (p-value =1.16 × 10-2). Since miR-144 is co-regulating with miR-130, miR-19 and miR-101 of rank 4 (p-value = 1.16 × 10-2) in our 4-term miRNA table, this suggests hsa-miR-144 may be neurodegenerative diseases related miRNA. Conclusions: This work identifies highly probable co-regulating miRNAs, which are refined from the prediction by computational tools using (1) signal-to-noise ratio to get high accurate regulating miRNAs for every gene, and (2) Gene Ontology to obtain functional related co-regulating miRNA groups. Our result has partly been supported by biological experiments. Based on prediction by TargetScanS, we found highly probable target gene groups in the Supplementary Information. This result might help biologists to find small set of miRNAs for genes of interest rather than huge amount of miRNA set. Supplementary Information:.

scholarly journals Symmetry Breaking in Tournaments

10.37236/3182 ◽  
2013 ◽  
Vol 20 (1) ◽  
Author(s):  
Antoni Lozano
Keyword(s):  
Symmetry Breaking ◽  
Upper Bounds ◽  
Minimum Size ◽  
Metric Dimension ◽  
Resolving Set ◽  
Nontrivial Automorphism ◽  
Determining Set ◽  
Strong Tournament

We provide upper bounds for the determining number and the metric dimension of tournaments. A set of vertices $S \subseteq V(T)$ is a determining set for a tournament $T$ if every nontrivial automorphism of $T$ moves at least one vertex of $S$, while $S$ is a resolving set for $T$ if every two distinct vertices in $T$ have different distances to some vertex in $S$. We show that the minimum size of a determining set for an order $n$ tournament (its determining number) is bounded by $\lfloor n/3 \rfloor$, while the minimum size of a resolving set for an order $n$ strong tournament (its metric dimension) is bounded by $\lfloor n/2 \rfloor$. Both bounds are optimal.

scholarly journals Regarding Two Conjectures on Clique and Biclique Partitions

10.37236/9564 ◽  
2021 ◽  
Vol 28 (4) ◽  
Author(s):  
Dhruv Rohatgi ◽  
John C. Urschel ◽  
Jake Wellens
Keyword(s):  
Upper Bounds ◽  
The Other ◽  
Lower And Upper Bounds ◽  
Minimum Number ◽  
Other Regarding

For a graph $G$, let $cp(G)$ denote the minimum number of cliques of $G$ needed to cover the edges of $G$ exactly once. Similarly, let $bp_k(G)$ denote the minimum number of bicliques (i.e. complete bipartite subgraphs of $G$) needed to cover each edge of $G$ exactly $k$ times. We consider two conjectures – one regarding the maximum possible value of $cp(G) + cp(\overline{G})$ (due to de Caen, Erdős, Pullman and Wormald) and the other regarding $bp_k(K_n)$ (due to de Caen, Gregory and Pritikin). We disprove the first, obtaining improved lower and upper bounds on $\max_G cp(G) + cp(\overline{G})$, and we prove an asymptotic version of the second, showing that $bp_k(K_n) = (1+o(1))n$.

scholarly journals Upper Bounds on the Minimum Size of Hamilton Saturated Hypergraphs

10.37236/5252 ◽  
2016 ◽  
Vol 23 (4) ◽  
Author(s):  
Andrzej Ruciński ◽  
Andrzej Żak
Keyword(s):  
Upper Bound ◽  
Upper Bounds ◽  
Minimum Size ◽  
Uniform Hypergraph ◽  
Open Case ◽  
Vertex Ordering

For $1\leqslant \ell< k$,  an $\ell$-overlapping $k$-cycle is a $k$-uniform hypergraph in which, for some cyclic vertex ordering, every edge consists of $k$ consecutive vertices and every two consecutive edges share exactly $\ell$ vertices.A $k$-uniform hypergraph $H$ is $\ell$-Hamiltonian saturated if $H$ does not contain an $\ell$-overlapping Hamiltonian $k$-cycle but every hypergraph obtained from $H$ by adding one edge does contain such a cycle. Let $\mathrm{sat}(n,k,\ell)$ be the smallest number of edges in an $\ell$-Hamiltonian saturated $k$-uniform hypergraph on $n$ vertices. In the case of graphs Clark and Entringer showed in 1983 that $\mathrm{sat}(n,2,1)=\lceil \tfrac{3n}2\rceil$. The present authors proved that for $k\geqslant 3$ and $\ell=1$, as well as for all $0.8k\leqslant \ell\leq k-1$, $\mathrm{sat}(n,k,\ell)=\Theta(n^{\ell})$. In this paper we prove two upper bounds which cover the remaining range of $\ell$. The first, quite technical one, restricted to $\ell\geqslant\frac{k+1}2$, implies in particular that for $\ell=\tfrac23k$ and $\ell=\tfrac34k$ we have $\mathrm{sat}(n,k,\ell)=O(n^{\ell+1})$. Our main result provides an upper bound $\mathrm{sat}(n,k,\ell)=O(n^{\frac{k+\ell}2})$ valid for all $k$ and $\ell$. In the smallest open case we improve it further to $\mathrm{sat}(n,4,2)=O(n^{\frac{14}5})$.

scholarly journals On the $P_3$-Hull Number of Kneser Graphs

10.37236/9903 ◽  
2021 ◽  
Vol 28 (3) ◽  
Author(s):  
Luciano N. Grippo ◽  
Adrián Pastine ◽  
Pablo Torres ◽  
Mario Valencia-Pabon ◽  
Juan C. Vera
Keyword(s):  
Upper Bounds ◽  
Minimum Size ◽  
Lower And Upper Bounds ◽  
Kneser Graph ◽  
Graph Homomorphisms ◽  
The Family ◽  
Vertex Set ◽  
Kneser Graphs

This paper considers an infection spreading in a graph; a vertex gets infected if at least two of its neighbors are infected. The $P_3$-hull number is the minimum size of a vertex set that eventually infects the whole graph. In the specific case of the Kneser graph $K(n,k)$, with $n\ge 2k+1$, an infection spreading on the family of $k$-sets of an $n$-set is considered. A set is infected whenever two sets disjoint from it are infected. We compute the exact value of the $P_3$-hull number of $K(n,k)$ for $n>2k+1$. For $n = 2k+1$, using graph homomorphisms from the Knesser graph to the Hypercube, we give lower and upper bounds.

Sign in / Sign up

Export Citation Format

Share Document