scholarly journals A Better Lower Bound on Average Degree of Online $k$-List-Critical Graphs

10.37236/6405 ◽  
2018 ◽  
Vol 25 (1) ◽  
Author(s):  
Landon Rabern
Keyword(s):  
Lower Bound ◽  
Average Degree ◽  
Critical Graphs ◽  
Critical Graph

We improve the best known bounds on average degree of online $k$-list-critical graphs for $k \geqslant 6$. Specifically, for $k \geqslant 7$ we show that every non-complete online $k$-list-critical graph has average degree at least $k-1 + \frac{(k-3)^2 (2 k-3)}{k^4-2 k^3-11 k^2+28 k-14}$ and every non-complete online $6$-list-critical graph has average degree at least $5 + \frac{93}{766}$. The same bounds hold for offline $k$-list-critical graphs.

scholarly journals A Better Lower Bound on Average Degree of 4-List-Critical Graphs

10.37236/5971 ◽  
2016 ◽  
Vol 23 (3) ◽  
Author(s):  
Landon Rabern
Keyword(s):  
Lower Bound ◽  
Short Note ◽  
Average Degree ◽  
Critical Graphs ◽  
Critical Graph

This short note proves that every non-complete $k$-list-critical graph has average degree at least $k-1 + \frac{k-3}{k^2-2k+2}$. This improves the best known bound for $k = 4,5,6$. The same bound holds for online $k$-list-critical graphs.

scholarly journals On Degree Properties of Crossing-Critical Families of Graphs

10.37236/7753 ◽  
2019 ◽  
Vol 26 (1) ◽  
Author(s):  
Drago Bokal ◽  
Mojca Bračič ◽  
Marek Derňár ◽  
Petr Hliněný
Keyword(s):  
Average Degree ◽  
Critical Graphs ◽  
Odd Degree ◽  
Vertex Degrees ◽  
Open Question

Answering an open question from 2007, we construct infinite $k$-crossing-critical families of graphs that contain vertices of any prescribed odd degree, for any sufficiently large $k$. To answer this question, we introduce several properties of infinite families of graphs and operations on the families allowing us to obtain new families preserving those properties. This conceptual setup allows us to answer general questions on behaviour of degrees in crossing-critical graphs: we show that, for any set of integers $D$ such that $\min(D)\geq 3$ and $3,4\in D$, and for any sufficiently large $k$, there exists a $k$-crossing-critical family such that the numbers in $D$ are precisely the vertex degrees that occur arbitrarily often in (large enough) graphs of this family. Furthermore, even if both $D$ and some average degree in the interval $(3,6)$ are prescribed, $k$-crossing-critical families exist for any sufficiently large $k$.

scholarly journals Split Domination Vertex Critical and Edge Critical Graphs

2020 ◽  
Vol 26 (1) ◽  
pp. 55-63
Author(s):  
Girish V R ◽  
Usha P
Keyword(s):  
Dominating Set ◽  
Domination Number ◽  
Minimum Cardinality ◽  
Critical Graphs ◽  
Critical Graph ◽  
Induced Graph ◽  
Vertex Removal ◽  
Edge Addition

A dominating set D of a graph G = (V;E) is a split dominating set ifthe induced graph hV 􀀀 Di is disconnected. The split domination number s(G)is the minimum cardinality of a split domination set. A graph G is called vertexsplit domination critical if s(G􀀀v) s(G) for every vertex v 2 G. A graph G iscalled edge split domination critical if s(G + e) s(G) for every edge e in G. Inthis paper, whether for some standard graphs are split domination vertex critical ornot are investigated and then characterized 2- ns-critical and 3- ns-critical graphswith respect to the diameter of a graph G with vertex removal. Further, it is shownthat there is no existence of s-critical graph for edge addition.

scholarly journals Minimal Weight in Union-Closed Families

10.37236/582 ◽  
2011 ◽  
Vol 18 (1) ◽  
Author(s):  
Victor Falgas-Ravry
Keyword(s):  
Lower Bound ◽  
Related Question ◽  
Average Degree ◽  
Additional Information ◽  
Minimal Weight ◽  
Multiplicative Factor ◽  
Finite Set

Let $\Omega$ be a finite set and let $\mathcal{S} \subseteq \mathcal{P}(\Omega)$ be a set system on $\Omega$. For $x\in \Omega$, we denote by $d_{\mathcal{S}}(x)$ the number of members of $\mathcal{S}$ containing $x$. A long-standing conjecture of Frankl states that if $\mathcal{S}$ is union-closed then there is some $x\in \Omega$ with $d_{\mathcal{S}}(x)\geq \frac{1}{2}|\mathcal{S}|$. We consider a related question. Define the weight of a family $\mathcal{S}$ to be $w(\mathcal{S}) := \sum_{A \in \mathcal{S}} |A|$. Suppose $\mathcal{S}$ is union-closed. How small can $w(\mathcal{S})$ be? Reimer showed $$w(\mathcal{S}) \geq \frac{1}{2} |\mathcal{S}| \log_2 |\mathcal{S}|,$$ and that this inequality is tight. In this paper we show how Reimer's bound may be improved if we have some additional information about the domain $\Omega$ of $\mathcal{S}$: if $\mathcal{S}$ separates the points of its domain, then $$w(\mathcal{S})\geq \binom{|\Omega|}{2}.$$ This is stronger than Reimer's Theorem when $\vert \Omega \vert > \sqrt{|\mathcal{S}|\log_2 |\mathcal{S}|}$. In addition we construct a family of examples showing the combined bound on $w(\mathcal{S})$ is tight except in the region $|\Omega|=\Theta (\sqrt{|\mathcal{S}|\log_2 |\mathcal{S}|})$, where it may be off by a multiplicative factor of $2$. Our proof also gives a lower bound on the average degree: if $\mathcal{S}$ is a point-separating union-closed family on $\Omega$, then $$ \frac{1}{|\Omega|} \sum_{x \in \Omega} d_{\mathcal{S}}(x) \geq \frac{1}{2} \sqrt{|\mathcal{S}| \log_2 |\mathcal{S}|}+ O(1),$$ and this is best possible except for a multiplicative factor of $2$.

The average degree of an edge-chromatic critical graph II

2007 ◽  
Vol 56 (3) ◽  
pp. 194-218 ◽  
Author(s):  
Douglas R. Woodall
Keyword(s):  
Average Degree ◽  
Critical Graph

On the minimal length of the longest trail in a fixed edge-density graph

2013 ◽  
Vol 11 (10) ◽  
Author(s):  
Vajk Szécsi
Keyword(s):  
Lower Bound ◽  
Minimal Length ◽  
Average Degree ◽  
Edge Density ◽  
Sharp Lower Bound

AbstractA nearly sharp lower bound on the length of the longest trail in a graph on n vertices and average degree k is given provided the graph is dense enough (k ≥ 12.5).

scholarly journals Characteristics of (γ,3)-critical graphs

2010 ◽  
Vol 4 (1) ◽  
pp. 197-206 ◽  
Author(s):  
D.A. Mojdeh ◽  
P.Y. Firoozi
Keyword(s):  
Edge Connectivity ◽  
Critical Graphs ◽  
Critical Graph

In this note the (?,3)-critical graphs are fairly classified. We show that a (?,k)- critical graph is not necessarily a (?,k?)-critical for k?? k and k,k? E {1,2,3}. The (2,3)-critical graphs are definitely characterized. Also the properties of (?,3)-critical graphs are verified once their edge connectivity are 3.

scholarly journals On the average degree of critical graphs with maximum degree six

2011 ◽  
Vol 311 (21) ◽  
pp. 2574-2576 ◽  
Author(s):  
Lianying Miao ◽  
Jibin Qu ◽  
Qingbo Sun
Keyword(s):  
Maximum Degree ◽  
Average Degree ◽  
Critical Graphs

The average degree of a subcubic edge‐chromatic critical graph

2018 ◽  
Vol 91 (2) ◽  
pp. 103-121
Author(s):  
Douglas R. Woodall
Keyword(s):  
Average Degree ◽  
Critical Graph
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