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).

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 On the Size of $(K_t,\mathcal{T}_k)$-Co-Critical Graphs

10.37236/8857 ◽  
2021 ◽  
Vol 28 (1) ◽  
Author(s):  
Zi-Xia Song ◽  
Jingmei Zhang
Keyword(s):  
Graph Theory ◽  
Complete Graph ◽  
Minimum Degree ◽  
Bootstrap Percolation ◽  
Critical Graphs ◽  
Critical Graph ◽  
The Family ◽  
Saturated Graphs ◽  
Minimum Number ◽  
Monochromatic Copy

Given an integer $r\geqslant 1$ and graphs $G, H_1, \ldots, H_r$, we write $G \rightarrow ({H}_1, \ldots, {H}_r)$ if every $r$-coloring of the edges of $G$ contains a monochromatic copy of $H_i$ in color $i$ for some $i\in\{1, \ldots, r\}$. A non-complete graph $G$ is $(H_1, \ldots, H_r)$-co-critical if $G \nrightarrow ({H}_1, \ldots, {H}_r)$, but $G+e\rightarrow ({H}_1, \ldots, {H}_r)$ for every edge $e$ in $\overline{G}$. In this paper, motivated by Hanson and Toft's conjecture [Edge-colored saturated graphs, J Graph Theory 11(1987), 191–196], we study the minimum number of edges over all $(K_t, \mathcal{T}_k)$-co-critical graphs on $n$ vertices, where $\mathcal{T}_k$ denotes the family of all trees on $k$ vertices. Following Day [Saturated graphs of prescribed minimum degree, Combin. Probab. Comput. 26 (2017), 201–207], we apply graph bootstrap percolation on a not necessarily $K_t$-saturated graph to prove that for all $t\geqslant4 $ and $k\geqslant\max\{6, t\}$, there exists a constant $c(t, k)$ such that, for all $n \ge (t-1)(k-1)+1$, if $G$ is a $(K_t, \mathcal{T}_k)$-co-critical graph on $n$ vertices, then $$ e(G)\geqslant \left(\frac{4t-9}{2}+\frac{1}{2}\left\lceil \frac{k}{2} \right\rceil\right)n-c(t, k).$$ Furthermore, this linear bound is asymptotically best possible when $t\in\{4,5\}$ and $k\geqslant6$. The method we develop in this paper may shed some light on attacking Hanson and Toft's conjecture.

scholarly journals Cliques in Graphs With Bounded Minimum Degree

2012 ◽  
Vol 21 (3) ◽  
pp. 457-482 ◽  
Author(s):  
ALLAN LO
Keyword(s):  
Minimum Degree ◽  
Extremal Graphs ◽  
Minimum Number

Let kr(n, δ) be the minimum number of r-cliques in graphs with n vertices and minimum degree at least δ. We evaluate kr(n, δ) for δ ≤ 4n/5 and some other cases. Moreover, we give a construction which we conjecture to give all extremal graphs (subject to certain conditions on n, δ and r).

On the double total dominator chromatic number of graphs

2021 ◽  
pp. 2150154
Author(s):  
Fairouz Beggas ◽  
Hamamache Kheddouci ◽  
Walid Marweni
Keyword(s):  
Chromatic Number ◽  
Minimum Degree ◽  
Domination Number ◽  
Vertex Coloring ◽  
Total Domination Number ◽  
Close Relationship ◽  
Minimum Number ◽  
Number Formula ◽  
Dominator Coloring ◽  
Proper Vertex

In this paper, we introduce and study a new coloring problem of graphs called the double total dominator coloring. A double total dominator coloring of a graph [Formula: see text] with minimum degree at least 2 is a proper vertex coloring of [Formula: see text] such that each vertex has to dominate at least two color classes. The minimum number of colors among all double total dominator coloring of [Formula: see text] is called the double total dominator chromatic number, denoted by [Formula: see text]. Therefore, we establish the close relationship between the double total dominator chromatic number [Formula: see text] and the double total domination number [Formula: see text]. We prove the NP-completeness of the problem. We also examine the effects on [Formula: see text] when [Formula: see text] is modified by some operations. Finally, we discuss the [Formula: see text] number of square of trees by giving some bounds.

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.

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 Minimum degree and the minimum size of K2t-saturated graphs

2007 ◽  
Vol 307 (9-10) ◽  
pp. 1108-1114 ◽  
Author(s):  
Ronald J. Gould ◽  
John R. Schmitt
Keyword(s):  
Minimum Degree ◽  
Minimum Size ◽  
Saturated Graphs

Inverse Sum Indeg Index of Subdivision, t-Subdivision Graphs, and Related Sums

2020 ◽  
pp. 104-119 ◽  
Author(s):  
Amitav Doley ◽  
Jibonjyoti Buragohain ◽  
A. Bharali
Keyword(s):  
Surface Area ◽  
Maximum Degree ◽  
Minimum Degree ◽  
Total Surface Area ◽  
Upper Bounds ◽  
Lower And Upper Bounds ◽  
Zagreb Index ◽  
Octane Isomers ◽  
Graph Parameters ◽  
The First Zagreb Index

The inverse sum indeg (ISI) index of a graph G is defined as the sum of the weights dG(u)dG(v)/dG(u)+dG(v) of all edges uv in G, where dG(u) is the degree of the vertex u in G. This index is found to be a significant predictor of total surface area of octane isomers. In this chapter, the authors present some lower and upper bounds for ISI index of subdivision graphs, t-subdivision graphs, s-sum and st -sum of graphs in terms of some graph parameters such as order, size, maximum degree, minimum degree, and the first Zagreb index. The extremal graphs are also characterized for their sharpness.

Sign in / Sign up

Export Citation Format

Share Document