scholarly journals On Domination in 2-Connected Cubic Graphs

10.37236/913 ◽  
2008 ◽  
Vol 15 (1) ◽  
Author(s):  
B. Y. Stodolsky
Keyword(s):  
Minimum Degree ◽  
Domination Number ◽  
The Other ◽  
Cubic Graphs ◽  
Cubic Graph ◽  
Other Hand ◽  
G 21 ◽  
Vertex Graph

In 1996, Reed proved that the domination number, $\gamma(G)$, of every $n$-vertex graph $G$ with minimum degree at least $3$ is at most $3n/8$ and conjectured that $\gamma(H)\leq\lceil n/3\rceil$ for every connected $3$-regular (cubic) $n$-vertex graph $H$. In [1] this conjecture was disproved by presenting a connected cubic graph $G$ on $60$ vertices with $\gamma(G)=21$ and a sequence $\{G_k\}_{k=1}^{\infty}$ of connected cubic graphs with $\lim_{k\to\infty}{\gamma(G_k)\over|V(G_k)|} \geq{1\over3}+{1\over69}$. All the counter-examples, however, had cut-edges. On the other hand, in [2] it was proved that $\gamma(G)\leq\ 4n/11$ for every connected cubic $n$-vertex graph $G$ with at least $10$ vertices. In this note we construct a sequence of graphs $\{G_k\}_{k=1}^{\infty}$ of $2$-connected cubic graphs with $\lim_{k\to\infty}{\gamma(G_k)\over|V(G_k)|} \geq{1\over3}+{1\over78}$, and a sequence $\{G_l'\}_{l=1}^{\infty}$ of connected cubic graphs where for each $G_l'$ we have ${\gamma(G_l')\over|V(G_l')|} >{1\over3}+{1\over69}$.

Bisections of Graphs Without Short Cycles

2017 ◽  
Vol 27 (1) ◽  
pp. 44-59 ◽  
Author(s):  
GENGHUA FAN ◽  
JIANFENG HOU ◽  
XINGXING YU
Keyword(s):  
Minimum Degree ◽  
The Other ◽  
Other Hand

Bollobás and Scott (Random Struct. Alg.21 (2002) 414–430) asked for conditions that guarantee a bisection of a graph with m edges in which each class has at most (1/4+o(1))m edges. We demonstrate that cycles of length 4 play an important role for this question. Let G be a graph with m edges, minimum degree δ, and containing no cycle of length 4. We show that if (i) G is 2-connected, or (ii) δ ⩾ 3, or (iii) δ ⩾ 2 and the girth of G is at least 5, then G admits a bisection in which each class has at most (1/4+o(1))m edges. We show that each of these conditions are best possible. On the other hand, a construction by Alon, Bollobás, Krivelevich and Sudakov shows that for infinitely many m there exists a graph with m edges and girth at least 5 for which any bisection has at least (1/4−o(1))m edges in one of the two classes.

scholarly journals On the Line Graph of the Zero Divisor Graph for the Ring of Gaussian Integers Modulo n

2012 ◽  
Vol 2012 ◽  
pp. 1-13 ◽  
Author(s):  
Khalida Nazzal ◽  
Manal Ghanem
Keyword(s):  
Zero Divisor ◽  
Line Graph ◽  
Domination Number ◽  
The Other ◽  
Graph Invariants ◽  
Gaussian Integers ◽  
Zero Divisor Graph ◽  
Other Hand

Let Γ(ℤn[i]) be the zero divisor graph for the ring of the Gaussian integers modulo n. Several properties of the line graph of Γ(ℤn[i]), L(Γ(ℤn[i])) are studied. It is determined when L(Γ(ℤn[i])) is Eulerian, Hamiltonian, or planer. The girth, the diameter, the radius, and the chromatic and clique numbers of this graph are found. In addition, the domination number of L(Γ(ℤn[i])) is given when n is a power of a prime. On the other hand, several graph invariants for Γ(ℤn[i]) are also determined.

scholarly journals The structure and the list 3-dynamic coloring of outer-1-planar graphs

2021 ◽  
Vol vol. 23, no. 3 (Graph Theory) ◽  
Author(s):  
Yan Li ◽  
Xin Zhang
Keyword(s):  
Planar Graph ◽  
Chromatic Number ◽  
Upper Bound ◽  
Local Structure ◽  
Planar Graphs ◽  
Minimum Degree ◽  
Outer Region ◽  
The Other ◽  
Structural Theorem ◽  
Other Hand

An outer-1-planar graph is a graph admitting a drawing in the plane so that all vertices appear in the outer region of the drawing and every edge crosses at most one other edge. This paper establishes the local structure of outer-1-planar graphs by proving that each outer-1-planar graph contains one of the seventeen fixed configurations, and the list of those configurations is minimal in the sense that for each fixed configuration there exist outer-1-planar graphs containing this configuration that do not contain any of another sixteen configurations. There are two interesting applications of this structural theorem. First of all, we conclude that every (resp. maximal) outer-1-planar graph of minimum degree at least 2 has an edge with the sum of the degrees of its two end-vertices being at most 9 (resp. 7), and this upper bound is sharp. On the other hand, we show that the list 3-dynamic chromatic number of every outer-1-planar graph is at most 6, and this upper bound is best possible.

scholarly journals Detection number of bipartite graphs and cubic graphs

2014 ◽  
Vol Vol. 16 no. 3 ◽  
Author(s):  
Frederic Havet ◽  
Nagarajan Paramaguru ◽  
Rathinaswamy Sampathkumar
Keyword(s):  
Positive Integer ◽  
Bipartite Graph ◽  
Connected Graph ◽  
Minimum Degree ◽  
Bipartite Graphs ◽  
Sufficient Condition ◽  
Cubic Graphs ◽  
Cubic Graph ◽  
International Audience ◽  
Np Complete

International audience For a connected graph G of order |V(G)| ≥3 and a k-labelling c : E(G) →{1,2,…,k} of the edges of G, the code of a vertex v of G is the ordered k-tuple (ℓ1,ℓ2,…,ℓk), where ℓi is the number of edges incident with v that are labelled i. The k-labelling c is detectable if every two adjacent vertices of G have distinct codes. The minimum positive integer k for which G has a detectable k-labelling is the detection number det(G) of G. In this paper, we show that it is NP-complete to decide if the detection number of a cubic graph is 2. We also show that the detection number of every bipartite graph of minimum degree at least 3 is at most 2. Finally, we give some sufficient condition for a cubic graph to have detection number 3.

scholarly journals Relaxed Two-Coloring of Cubic Graphs

2005 ◽  
Vol DMTCS Proceedings vol. AE,... (Proceedings) ◽  
Author(s):  
Robert Berke ◽  
Tibor Szabó
Keyword(s):  
Regular Graph ◽  
Maximum Degree ◽  
Independent Set ◽  
The Other ◽  
Cubic Graphs ◽  
Other Hand ◽  
Color Class ◽  
International Audience

International audience We show that any graph of maximum degree at most $3$ has a two-coloring, such that one color-class is an independent set while the other color induces monochromatic components of order at most $189$. On the other hand for any constant $C$ we exhibit a $4$-regular graph, such that the deletion of any independent set leaves at least one component of order greater than $C$. Similar results are obtained for coloring graphs of given maximum degree with $k+ \ell$ colors such that $k$ parts form an independent set and $\ell$ parts span components of order bounded by a constant. A lot of interesting questions remain open.

scholarly journals Exponential Domination in Subcubic Graphs

10.37236/5711 ◽  
2016 ◽  
Vol 23 (4) ◽  
Author(s):  
Stéphane Bessy ◽  
Pascal Ochem ◽  
Dieter Rautenbach
Keyword(s):  
Dominating Set ◽  
Domination Number ◽  
Discrete Math ◽  
Cubic Graphs ◽  
Cubic Graph ◽  
Minimum Order ◽  
Natural Variant ◽  
Dominating Set Problem ◽  
Subcubic Graphs ◽  
Domination In Graphs

As a natural variant of domination in graphs, Dankelmann et al. [Domination with exponential decay, Discrete Math. 309 (2009) 5877-5883] introduced exponential domination, where vertices are considered to have some dominating power that decreases exponentially with the distance, and the dominated vertices have to accumulate a sufficient amount of this power emanating from the dominating vertices. More precisely, if $S$ is a set of vertices of a graph $G$, then $S$ is an exponential dominating set of $G$ if $\sum\limits_{v\in S}\left(\frac{1}{2}\right)^{{\rm dist}_{(G,S)}(u,v)-1}\geq 1$ for every vertex $u$ in $V(G)\setminus S$, where ${\rm dist}_{(G,S)}(u,v)$ is the distance between $u\in V(G)\setminus S$ and $v\in S$ in the graph $G-(S\setminus \{ v\})$. The exponential domination number $\gamma_e(G)$ of $G$ is the minimum order of an exponential dominating set of $G$.In the present paper we study exponential domination in subcubic graphs. Our results are as follows: If $G$ is a connected subcubic graph of order $n(G)$, then $$\frac{n(G)}{6\log_2(n(G)+2)+4}\leq \gamma_e(G)\leq \frac{1}{3}(n(G)+2).$$ For every $\epsilon>0$, there is some $g$ such that $\gamma_e(G)\leq \epsilon n(G)$ for every cubic graph $G$ of girth at least $g$. For every $0<\alpha<\frac{2}{3\ln(2)}$, there are infinitely many cubic graphs $G$ with $\gamma_e(G)\leq \frac{3n(G)}{\ln(n(G))^{\alpha}}$. If $T$ is a subcubic tree, then $\gamma_e(T)\geq \frac{1}{6}(n(T)+2).$ For a given subcubic tree, $\gamma_e(T)$ can be determined in polynomial time. The minimum exponential dominating set problem is APX-hard for subcubic graphs.

scholarly journals Cycle Lengths Modulo k in Large 3-connected Cubic Graphs, Advances in Combinatorics

2021 ◽  
Author(s):  
Kasper S. Lyngsie ◽  
Martin Merker
Keyword(s):  
Graph Theory ◽  
Minimum Degree ◽  
Average Degree ◽  
Cubic Graphs ◽  
Cubic Graph ◽  
Open Problems ◽  
The Third ◽  
Long Line ◽  
Cycle Lengths ◽  
Parity Condition

The existence of cycles with a given length is classical topic in graph theory with a plethora of open problems. Examples related to the main result of this paper include a conjecture of Burr and Erdős from 1976 asked whether for every integer $m$ and a positive odd integer $k$, there exists $d$ such that every graph with average degree at least $d$ contains a cycle of length $m$ modulo $k$; this conjecture was proven by Bollobás in [Bull. London Math. Soc. 9 (1977), 97-98]( https://doi.org/10.1112/blms/9.1.97). Another example is a problem of Erdős from the 1990s asking whether there exists $A\subseteq\mathbb{N}$ with zero density and constants $n_0$ and $d_0$ such that every graph with at least $n_0$ vertices and the average degree at least $d_0$ contains a cycle with length in the set $A$, which was resolved by Verstraete in [J. Graph Theory 49 (2005), 151-167]( https://doi.org/10.1002/jgt.20072). In 1983, Thomassen conjectured that for all integers $m$ and $k$, every graph with minimum degree $k+1$ contains a cycle of length $2m$ modulo $k$. Note that the parity condition in the first and the third conjectures is necessary because of bipartite graphs. The current paper contributes to this long line of research by proving that for every integer $m$ and a positive odd integer $k$, every sufficiently large $3$-connected cubic graph contains a cycle of length $m$ modulo $k$. The result is the best possible in the sense that the same conclusion is not true for $2$-connected cubic graphs or $3$-connected graphs with minimum degree three.

2-domination dot-stable and dot-critical graphs

2019 ◽  
pp. 2150010
Author(s):  
Karima Attalah ◽  
Mustapha Chellali
Keyword(s):  
Dominating Set ◽  
Domination Number ◽  
The Other ◽  
Minimum Cardinality ◽  
Critical Graphs ◽  
Other Hand

A set [Formula: see text] of vertices in a graph [Formula: see text] is a [Formula: see text]-dominating set of [Formula: see text] if every vertex of [Formula: see text] is adjacent to at least two vertices in [Formula: see text] The [Formula: see text]-domination number is the minimum cardinality of a [Formula: see text]-dominating set of [Formula: see text]. A graph is [Formula: see text]-domination dot-stable if the contraction of any arbitrary edge has no effect on the [Formula: see text]-domination number. On the other hand, a graph is [Formula: see text]-domination dot-critical if the contraction of any arbitrary edge decreases the [Formula: see text]-domination number. We present some properties of these graphs and we provide a characterization of all trees that are [Formula: see text]-domination dot-stable or [Formula: see text]-domination dot-critical.

A study of cometary eruptions through OH radio-lines

1999 ◽  
Vol 173 ◽  
pp. 249-254
Author(s):  
A.M. Silva ◽  
R.D. Miró
Keyword(s):  
Short Duration ◽  
Growth Curves ◽  
The Other ◽  
Initial Mass ◽  
Radio Lines ◽  
Other Hand ◽  
Sudden Rise

AbstractWe have developed a model for theH2OandOHevolution in a comet outburst, assuming that together with the gas, a distribution of icy grains is ejected. With an initial mass of icy grains of 108kg released, theH2OandOHproductions are increased up to a factor two, and the growth curves change drastically in the first two days. The model is applied to eruptions detected in theOHradio monitorings and fits well with the slow variations in the flux. On the other hand, several events of short duration appear, consisting of a sudden rise ofOHflux, followed by a sudden decay on the second day. These apparent short bursts are frequently found as precursors of a more durable eruption. We suggest that both of them are part of a unique eruption, and that the sudden decay is due to collisions that de-excite theOHmaser, when it reaches the Cometopause region located at 1.35 × 105kmfrom the nucleus.

Closing the GAP—A 5Å Scanning Microscope

1969 ◽  
Vol 27 ◽  
pp. 6-7
Author(s):  
A. V. Crewe
Keyword(s):  
Normal Form ◽  
The Other ◽  
Resolving Power ◽  
Scanning Microscope ◽  
Conventional Microscope ◽  
Other Hand ◽  
Closing The Gap ◽  
Thermal Sources ◽  
Better Than ◽  
Transmission Microscope

We have become accustomed to differentiating between the scanning microscope and the conventional transmission microscope according to the resolving power which the two instruments offer. The conventional microscope is capable of a point resolution of a few angstroms and line resolutions of periodic objects of about 1Å. On the other hand, the scanning microscope, in its normal form, is not ordinarily capable of a point resolution better than 100Å. Upon examining reasons for the 100Å limitation, it becomes clear that this is based more on tradition than reason, and in particular, it is a condition imposed upon the microscope by adherence to thermal sources of electrons.

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