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A new lower bound on the number of edges in colour-critical graphs and hypergraphs

Alexandr V. Kostochka, Michael Stiebitz
2003 Journal of combinatorial theory. Series B (Print)  
A graph G is called k-critical if it has chromatic number k; but every proper subgraph of G is ðk À 1Þ-colourable.  ...  We prove that every k-critical graph ðkX6Þ on nXk þ 2 vertices has at least 1 2 ðk À 1 þ kÀ3 ðkÀcÞðkÀ1ÞþkÀ3 Þn edges where c ¼ ðk À 5Þð 1 2 À 1 ðkÀ1ÞðkÀ2Þ Þ: This improves earlier bounds established by  ...  Critical graphs were first defined and used by Dirac [5] in 1951. In the present paper a new lower bound for the number of edges in a k-critical graph on n vertices is established.  ... 
doi:10.1016/s0095-8956(02)00035-7 fatcat:ulq2ciua5veova3bimygtoqgty

On the Number of Edges in Hypergraphs Critical with Respect to Strong Colourings

Alexandr V. Kostochka, Douglas R. Woodall
2000 European journal of combinatorics (Print)  
In the case when the generated graph of the hypergraph has bounded clique number, we give a lower bound that is valid for sufficiently large k and is asymptotically tight in k; this bound also holds for  ...  We estimate the minimum number of edges possible in a k-critical t-uniform hypergraph with a given number of vertices.  ...  The work of this author was also partly supported by grants 96-01-01614 and 97-01-01075 of the Russian Foundation for Fundamental Research.  ... 
doi:10.1006/eujc.1999.0330 fatcat:t5f2ncet4vhgrmkmjdezzynmle

Page 2606 of Mathematical Reviews Vol. , Issue 96e [page]

1996 Mathematical Reviews  
Summary: “A Hoang graph G = (V,E) is a graph whose edges may be coloured with two colours (red and white) such that no induced path on four vertices has wings of the same colour.  ...  The author introduces the notion of a co-edge in a hypergraph and defines what is meant by a free coloring. The co-edges in a free coloring contain at least two vertices of the same color.  ... 

Connected τ -critical hypergraphs of minimal size

Matěj Stehlík
2005 Discrete Mathematics & Theoretical Computer Science  
International audience A hypergraph $\mathscr{H}$ is $τ$ -critical if $τ (\mathscr{H}-E) < τ (\mathscr{H})$ for every edge $E ∈\mathscr{H}$, where $τ (\mathscr{H})$ denotes the transversal number of $\  ...  It can be shown that a connected $τ$ -critical hypergraph $\mathscr{H}$ has at least $2τ (\mathscr{H})-1$ edges; this generalises a classical theorem of Gallai on $χ$ -vertex-critical graphs with connected  ...  We first present a sharp lower bound on the number of edges in a connected τ -critical hypergraph, and then investigate the cases where equality is attained.  ... 
doi:10.46298/dmtcs.3397 fatcat:xzevnh6uvzdb5jkasffq24jfqa

Page 2334 of Mathematical Reviews Vol. , Issue 82f [page]

1982 Mathematical Reviews  
He also establishes an analogous result for hypergraphs and gives best possible lower bounds in certain special cases. 82f:05030 J. W. Moon (Edmonton, Alta.)  ...  Given a cellularly imbedded graph L—S and an assignment of voltages to the edges of L in a group, one may construct a branched covering p:S—S of surfaces, such that the graph p™'(L) is a covering graph  ... 

Path Coupling Using Stopping Times [chapter]

Magnus Bordewich, Martin Dyer, Marek Karpinski
2005 Lecture Notes in Computer Science  
We show that the Glauber dynamics for independent sets in a hypergraph mixes rapidly as long as the maximum degree ∆ of a vertex and the minimum size m of an edge satisfy m ≥ 2∆ + 1.  ...  We also state results that the Glauber dynamics for proper q-colourings of a hypergraph mixes rapidly if m ≥ 4 and q > ∆, and if m = 3 and q ≥ 1.65∆.  ...  Acknowledgments We are grateful to Tom Hayes for commenting on an earlier draft of this paper, and to Mary Cryan for useful discussions at an early stage of this work.  ... 
doi:10.1007/11537311_3 fatcat:nhgjsh2c3nftllhewoagseffru

Page 6546 of Mathematical Reviews Vol. , Issue 2003i [page]

2003 Mathematical Reviews  
The main feature is that mixed hypergraphs represent structures in which problems on both the minimum and maximum number of colours occur.  ...  It belongs on the shelves of everyone who works not only in graph and hypergraph theory, but more generally in discrete mathematics.  ... 

Page 2347 of Mathematical Reviews Vol. , Issue 2003d [page]

2003 Mathematical Reviews  
A hypergraph H is said to be a hypergraph with p-definable edges if any edge of H contains at least p+ 1 vertices and any p vertices of H are contained in not more than one edge.  ...  A graph is 3-domination-critical if its domination number is 3 and the addition of any edge reduces the domination number to 2.  ... 

On the minimum degree of minimal Ramsey graphs for multiple colours [article]

Jacob Fox, Andrey Grinshpun, Anita Liebenau, Yury Person, Tibor Szabo
2015 arXiv   pre-print
A graph G is r-Ramsey for a graph H, denoted by G→ (H)_r, if every r-colouring of the edges of G contains a monochromatic copy of H.  ...  We also give an upper bound on s_r(K_k) which is polynomial in both r and k, and we determine s_r(K_3) up to a factor of log r.  ...  Therefore, the lower bound on s r (K 3 ) does not necessarily imply a similar lower bound on s r (K k ). We can in fact only prove a super-quadratic lower bound on s r (K k ) that is slightly weaker.  ... 
arXiv:1502.02881v1 fatcat:djpcaevlunafhabm47tfunkkvu

On the minimum degree of minimal Ramsey graphs for multiple colours

Jacob Fox, Andrey Grinshpun, Anita Liebenau, Yury Person, Tibor Szabó
2016 Journal of combinatorial theory. Series B (Print)  
A graph G is r-Ramsey for a graph H, denoted by G → (H) r , if every r-colouring of the edges of G contains a monochromatic copy of H.  ...  We also give an upper bound on s r (K k ) which is polynomial in both r and k, and we determine s r (K 3 ) up to a factor of log r.  ...  Therefore, the lower bound on s r (K 3 ) does not necessarily imply a similar lower bound on s r (K k ). We can in fact only prove a super-quadratic lower bound on s r (K k ) that is slightly weaker.  ... 
doi:10.1016/j.jctb.2016.03.006 fatcat:jjybf7twvvfknixguq6pweh6ni

Rainbow Turán Problems

PETER KEEVASH, DHRUV MUBAYI, BENNY SUDAKOV, JACQUES VERSTRAËTE
2006 Combinatorics, probability & computing  
For a fixed graph H, we define the rainbow Turán number ex * (n, H) to be the maximum number of edges in a graph on n vertices that has a proper edge-colouring with no rainbow H.  ...  Recall that the (ordinary) Turán number ex(n, H) is the maximum number of edges in a graph on n vertices that does not contain a copy of H.  ...  Thus a bound on the number edges in a graph with no rainbow C 2k gives a bound on the size of a B * k -set. A related and more commonly studied condition is the following.  ... 
doi:10.1017/s0963548306007760 fatcat:wap4bn3jt5dw3mq4tvjqem7kdy

Covering graphs by monochromatic trees and Helly-type results for hypergraphs [article]

Matija Bucić, Dániel Korándi, Benny Sudakov
2020 arXiv   pre-print
How many monochromatic paths, cycles or general trees does one need to cover all vertices of a given r-edge-coloured graph G?  ...  Roughly speaking, this question asks for the maximum number of vertices needed to cover all the edges of a hypergraph H if it is known that any collection of a few edges of H has a small cover.  ...  In particular, we are grateful for a suggestion on how to rewrite the proof of Lemma 4.3 to make it easier to follow.  ... 
arXiv:1902.05055v4 fatcat:ljact63gufg23cbddedlwnki4q

Path Coupling Using Stopping Times and Counting Independent Sets and Colourings in Hypergraphs [article]

Magnus Bordewich, Martin Dyer, Marek Karpinski
2005 arXiv   pre-print
We show that the Glauber dynamics for independent sets in a hypergraph mixes rapidly as long as the maximum degree Delta of a vertex and the minimum size m of an edge satisfy m>= 2Delta+1.  ...  We also show that the Glauber dynamics for proper q-colourings of a hypergraph mixes rapidly if m>= 4 and q > Delta, and if m=3 and q>=1.65Delta.  ...  Acknowledgments We are grateful to Tom Hayes for commenting on an earlier draft of this paper, and to Mary Cryan for useful discussions at an early stage of this work.  ... 
arXiv:math/0501081v2 fatcat:4sjn6ofyvbdbjf5cphysail6k4

The colour theorems of Brooks and Gallai extended

A.V. Kostochka, M. Stiebitz, B. Wirth
1996 Discrete Mathematics  
One of the basic results in graph colouring is Brooks' theorem [-4] which asserts that the chromatic number of every connected graph, that is not a complete graph or an odd cycle, does not exceed its maximum  ...  In this note, we use a reduction idea in order to give a new short proof of this result and to extend it to hypergraphs.  ...  For colour-critical graphs the above Theorem was proved by Gallai [6] . Moreover, using his result, Gallai established a lower bound for the number of edges of a k-colour-critical graph.  ... 
doi:10.1016/0012-365x(95)00294-7 fatcat:rztulceyejee7i4mqxxcivkqsy

Page 4111 of Mathematical Reviews Vol. , Issue 98G [page]

1998 Mathematical Reviews  
(SGP-SING; Singapore) On graphs in which any pair of colour classes but one induces a tree.  ...  They then establish bounds on the size and maximum degree of a connected graph in terms of its order and diameter and show that these same bounds are sharp for L-graphs as well.  ... 
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