scholarly journals On-line List Colouring of Random Graphs

10.37236/5003 ◽  
2015 ◽  
Vol 22 (2) ◽  
Author(s):  
Alan Frieze ◽  
Dieter Mitsche ◽  
Xavier Pérez-Giménez ◽  
Paweł Prałat
Keyword(s):  
Chromatic Number ◽  
Random Graphs ◽  
Constant Factor ◽  
Average Degree ◽  
Multiplicative Constant ◽  
Line List ◽  
Multiplicative Factor ◽  
On Line ◽  
Choice Number

In this paper, the on-line list colouring of binomial random graphs $\mathcal{G}(n,p)$ is studied. We show that the on-line choice number of $\mathcal{G}(n,p)$ is asymptotically almost surely asymptotic to the chromatic number of $\mathcal{G}(n,p)$, provided that the average degree $d=p(n-1)$ tends to infinity faster than $(\log \log n)^{1/3} (\log n)^2 n^{2/3}$. For sparser graphs, we are slightly less successful; we show that if $d \ge (\log n)^{2+\epsilon}$ for some $\epsilon>0$, then the on-line choice number is larger than the chromatic number by at most a multiplicative factor of $C$, where $C \in [2,4]$, depending on the range of $d$. Also, for $d=O(1)$, the on-line choice number is by at most a multiplicative constant factor larger than the chromatic number.

A General Upper Bound on the List Chromatic Number of Locally Sparse Graphs

2002 ◽  
Vol 11 (1) ◽  
pp. 103-111 ◽  
Author(s):  
VAN H. VU
Keyword(s):  
Chromatic Number ◽  
Maximum Degree ◽  
Present Author ◽  
Upper Bounds ◽  
Constant Factor ◽  
Chromatic Index ◽  
Sparse Graphs ◽  
Critical Graphs ◽  
Multiplicative Constant ◽  
List Chromatic Number

Suppose that G is a graph with maximum degree d(G) such that, for every vertex v in G, the neighbourhood of v contains at most d(G)2/f (f > 1) edges. We show that the list chromatic number of G is at most Kd(G)/log f, for some positive constant K. This result is sharp up to the multiplicative constant K and strengthens previous results by Kim [9], Johansson [7], Alon, Krivelevich and Sudakov [3], and the present author [18]. This also motivates several interesting questions.As an application, we derive several upper bounds for the strong (list) chromatic index of a graph, under various assumptions. These bounds extend earlier results by Faudree, Gyárfás, Schelp and Tuza [6] and Mahdian [13] and determine, up to a constant factor, the strong (list) chromatic index of a random graph. Another application is an extension of a result of Kostochka and Steibitz [10] concerning the structure of list critical graphs.

scholarly journals On-Line List Colouring of Graphs

10.37236/216 ◽  
2009 ◽  
Vol 16 (1) ◽  
Author(s):  
Xuding Zhu
Keyword(s):  
Bipartite Graph ◽  
Polynomial Time ◽  
Complete Bipartite Graph ◽  
Polynomial Time Algorithm ◽  
Time Algorithm ◽  
Line List ◽  
Open Questions ◽  
On Line ◽  
Choice Number

This paper studies on-line list colouring of graphs. It is proved that the on-line choice number of a graph $G$ on $n$ vertices is at most $\chi(G) \ln n+1$, and the on-line $b$-choice number of $G$ is at most ${e\chi(G)-1\over e-1} (b-1+ \ln n)+b$. Suppose $G$ is a graph with a given $\chi(G)$-colouring of $G$. Then for any $(\chi(G) \ln n +1)$-assignment $L$ of $G$, we give a polynomial time algorithm which constructs an $L$-colouring of $G$. For any $({e\chi(G)-1\over e-1} (b-1+ \ln n)+b)$-assignment $L$ of $G$, we give a polynomial time algorithm which constructs an $(L,b)$-colouring of $G$. We then characterize all on-line $2$-choosable graphs. It is also proved that a complete bipartite graph of the form $K_{3,q}$ is on-line $3$-choosable if and only if it is $3$-choosable, but there are graphs of the form $K_{6,q}$ which are $3$-choosable but not on-line $3$-choosable. Some open questions concerning on-line list colouring are posed in the last section.

scholarly journals On-Line List Colouring of Random Graphs

2017 ◽  
pp. 47-53
Author(s):  
Alan Frieze ◽  
Dieter Mitsche ◽  
Xavier Pérez-Giménez ◽  
Paweł Prałat
Keyword(s):  
Random Graphs ◽  
Line List ◽  
On Line

scholarly journals Mr. Paint and Mrs. Correct go Fractional

10.37236/627 ◽  
2011 ◽  
Vol 18 (1) ◽  
Author(s):  
Grzegorz Gutowski
Keyword(s):  
Chromatic Number ◽  
The Other ◽  
Line List ◽  
Fractional Chromatic Number ◽  
Other Hand ◽  
On Line

We study a fractional counterpart of the on-line list colouring game "Mr. Paint and Mrs. Correct" introduced recently by Schauz. We answer positively a question of Zhu by proving that for any given graph the on-line choice ratio and the (off-line) choice ratio coincide. On the other hand it is known from the paper of Alon et al. that the choice ratio equals the fractional chromatic number. It was also shown that the limits used in the definitions of these last two notions can be realised. We show that this is not the case for the on-line choice ratio. Both our results are obtained by exploring the strong links between the on-line choice ratio, and a new on-line game with probabilistic flavour which we introduce.

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 $2$.

On the Chromatic Number of Random Graphs

2007 ◽  
pp. 777-788 ◽  
Author(s):  
Amin Coja-Oghlan ◽  
Konstantinos Panagiotou ◽  
Angelika Steger
Keyword(s):  
Chromatic Number ◽  
Random Graphs

scholarly journals On Komlós’ tiling theorem in random graphs

2019 ◽  
Vol 29 (1) ◽  
pp. 113-127
Author(s):  
Rajko Nenadov ◽  
Nemanja Škorić
Keyword(s):  
Chromatic Number ◽  
Random Graph ◽  
Random Graphs ◽  
Upper Bound ◽  
Minimum Degree ◽  
Arbitrary Graph ◽  
Spanning Subgraph ◽  
Delta G ◽  
Image Position ◽  
Vertex Disjoint

AbstractGiven graphs G and H, a family of vertex-disjoint copies of H in G is called an H-tiling. Conlon, Gowers, Samotij and Schacht showed that for a given graph H and a constant γ>0, there exists C>0 such that if $p \ge C{n^{ - 1/{m_2}(H)}}$ , then asymptotically almost surely every spanning subgraph G of the random graph 𝒢(n, p) with minimum degree at least $\delta (G) \ge (1 - \frac{1}{{{\chi _{{\rm{cr}}}}(H)}} + \gamma )np$ contains an H-tiling that covers all but at most γn vertices. Here, χcr(H) denotes the critical chromatic number, a parameter introduced by Komlós, and m2(H) is the 2-density of H. We show that this theorem can be bootstrapped to obtain an H-tiling covering all but at most $\gamma {(C/p)^{{m_2}(H)}}$ vertices, which is strictly smaller when $p \ge C{n^{ - 1/{m_2}(H)}}$ . In the case where H = K3, this answers the question of Balogh, Lee and Samotij. Furthermore, for an arbitrary graph H we give an upper bound on p for which some leftover is unavoidable and a bound on the size of a largest H -tiling for p below this value.

scholarly journals Application of polynomial method to on-line list colouring of graphs

2012 ◽  
Vol 33 (5) ◽  
pp. 872-883 ◽  
Author(s):  
Po-Yi Huang ◽  
Tsai-Lien Wong ◽  
Xuding Zhu
Keyword(s):  
Polynomial Method ◽  
Line List ◽  
On Line

scholarly journals The set chromatic number of random graphs

2016 ◽  
Vol 215 ◽  
pp. 61-70 ◽  
Author(s):  
Andrzej Dudek ◽  
Dieter Mitsche ◽  
Paweł Prałat
Keyword(s):  
Chromatic Number ◽  
Random Graphs
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