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Gallai theorems for graphs, hypergraphs, and set systems

1988
*
Discrete Mathematics
*

Acknowledgement The authors thank the referee

doi:10.1016/0012-365x(88)90192-6
fatcat:btjc5gs7fnfajdr737tn7b4xba
*for*valuable comments which led to the*Theorem*11. ...*Gallai**theorems**for*hereditary*set**systems*In this section we will prove a general*theorem*concerning hereditary properties of*sets*, from which one can obtain as corollaries a variety of*Gallai**theorems*... As our last illustration of*Theorem*2, we let S = V(H), the vertex*set*of a*hypergraph*H with edge*set*8. ...##
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Connected τ -critical hypergraphs of minimal size

2005
*
Discrete Mathematics & Theoretical Computer Science
*

It can be shown that a connected $τ$ -critical

doi:10.46298/dmtcs.3397
fatcat:xzevnh6uvzdb5jkasffq24jfqa
*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 ... 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 $\ ...*Theorem*8 (*Gallai*1963 ) A χ-vertex-critical*graph*G with a connected complement has at least 2χ(G)− 1 vertices. ...##
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Extensions of Gallai's graph covering theorems for uniform hypergraphs

1991
*
Journal of combinatorial theory. Series B (Print)
*

Thus, rk" 6 rf" + (r -1) f', r(k" -f") < (r -1)f' < (r -l)p,

doi:10.1016/0095-8956(91)90094-z
fatcat:moh5xtbuzjd4zagy3m34qnfzdu
*and*the*theorem*follows. 1 Before verifying*Theorems*2 ment concerning the matching*and*3, we prove the following number of*set**systems*. ... Then a result of [2] states the following:*THEOREM*A.*For*every*graph*2, v(X) + p(X) = 1x1. In fact,*Gallai*proved this statement in a much stronger form:*THEOREM*B. ...##
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Motivations and history of some of my conjectures

1997
*
Discrete Mathematics
*

If H is the

doi:10.1016/s0012-365x(96)00161-6
fatcat:weasdrfjlzh75fi5din2g2uyzm
*hypergraph*dual of a Steiner triple*system*, the same bound*for*q(H) was also conjectured by J. Colbourn*and*M. Colbourn [24]. ... Thus, a*graph*G is a linear*hypergraph*,*and*Vizing's*theorem*is equivalent to: q(GC) = A(G'). ... Faber*and*L. Lovasz., Open Problem, in: Berge ...##
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Minimum number of elements representing a set system of given rank

1989
*
Journal of combinatorial theory. Series A
*

Answering a 25-year-old problem of Erdos

doi:10.1016/0097-3165(89)90064-2
fatcat:wrhrevzwejadbf5z2qv6tlmz7e
*and**Gallai*, we prove that if a*set**system*Z of rank r cannot be represented by I elements, then there is a subsystem HO' s Z on less than ( :I! ... Moreover, we determine the maximum cardinality of strongly independent vertex*sets*in r-critical*and*intersecting v-critical*hypergraphs*of given rank,*and*describe the extremal structures. ... Gyirfk*and*J. Lehel*for*stimulating discussions*and**for*their comments on a preliminary version of this note. ...##
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Coloring hypergraphs of low connectivity
[article]

2018
*
arXiv
*
pre-print

*For*k ≥ 4, the family H_k is the smallest that contains all complete

*graphs*K_k+1

*and*is closed under Hajós joins.

*For*the proofs of the above results we use critical

*hypergraphs*. ...

*For*a

*hypergraph*G, let χ(G), Δ(G),

*and*λ(G) denote the chromatic number, the maximum degree,

*and*the maximum local edge connectivity of G, respectively. ... Next to the Hajós construction there is another construction

*for*critical

*hypergraphs*, first used by Dirac

*for*critical

*graphs*(see

*Gallai*[7, (2.1)]). ...

##
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Page 3400 of Mathematical Reviews Vol. , Issue 86h
[page]

1986
*
Mathematical Reviews
*

From the text: “We strengthen classical

*theorems*of Petersen, Ore, Babler,*and**Gallai*on the existence of regular factors in regular*graphs*to nearly regular*graphs*. ... We prove the following two*theorems*: (i) a*graph*G is [2a, 2b]-factorable if*and*only if G is a (2am, 2bm|-*graph**for*some integer m,*and*(ii) every [8m + 2k, 10m + 2k]-*graph*is [1, 2]-factorable.” ...##
###
Extension of paths and cycles for hypergraphs

2014
*
Electronic Notes in Discrete Mathematics
*

There are many results

doi:10.1016/j.endm.2013.11.002
fatcat:eqpa2dufhvb7tpykndmnsrzea4
*for**hypergraphs*that uses the traditional path*and*cycle definition of Berge, so almost all of them can be considered with the new definition. ... In [5] we defined the hamiltonian cycle in*hypergraphs*in a new way. The definition can be extended to paths*and*cycles as well. ...*Theorem*3.1 (Erdős-*Gallai*[1] ) Let G be a*graph*on n vertices containing no path of length k. Then e(G) ≤ 1 2 (k−1)n. Equality holds iff G is the disjoint union of complete*graphs*on k vertices. ...##
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Color-Critical Graphs and Hypergraphs with Few Edges: A Survey
[chapter]

2006
*
Bolyai Society Mathematical Studies
*

The current situation with bounds on the smallest number of edges in colorcritical

doi:10.1007/978-3-540-32439-3_9
fatcat:6hmpcen2c5godpbueoyqnigzye
*graphs**and**hypergraphs*is discussed. ... A partial (r, l)-*system*is an r-uniform*hypergraph*in which every*set*of l vertices is contained in at most one edge. ...*For*example, a*theorem*of*Gallai*(described in Section 4 below) says that E(G) / V (G) ≥ 0.5(k − 1 + k−3 k 2 −3 ) when G is a kcritical*graph*other than K k . ...##
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Page 2292 of Mathematical Reviews Vol. , Issue 87e
[page]

1987
*
Mathematical Reviews
*

Let G be a (p,q)-

*graph*, i.e. a*graph*with p vertices*and*gq edges, whose edge*set*E(G) has been assigned a specific ordering. ... {*For*the entire collection see MR 86g:05026.} Lars Dgvling Andersen (Aalborg) Koester, G. (DDR-MLU) 87e:05071 Note to a problem of T.*Gallai**and*G. A. Dirac. Combinatorica 5 (1985), no. 3, 227-228. ...##
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Color-critical Graphs and Hereditary Hypergraphs
[article]

2019
*
arXiv
*
pre-print

A quick proof of Gallai's celebrated

arXiv:1910.11302v1
fatcat:zjv764nrmvexhhtzdcxv7usr2m
*theorem*on color-critical*graphs*is given from Gallai's simple, ingenious lemma on factor-critical*graphs*, in terms of partitioning the vertex-*set*into a minimum number ... We then show examples of applying the results to new problems*and*indicate the way to algorithms*and*refined complexity results*for*all these examples at the same time. ... Fix a (not necessarily finite)*set*G of (di)*graphs**and**for*each (di)*graph*When are the*Theorem*or its corollaries meaningful or even interesting*for*H(G, G)? ...##
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Avoiding long Berge cycles
[article]

2018
*
arXiv
*
pre-print

We also show that the bound is attained only

arXiv:1805.04195v2
fatcat:7pbywh44h5g6rgb4dvqoa5kvpy
*for*connected r-uniform*hypergraphs*in which every block is the complete*hypergraph*K^(r)_k-1. ... Let n≥ k≥ r+3*and*H be an n-vertex r-uniform*hypergraph*. We show that if | H|> n-1/k-2k-1r then H contains a Berge cycle of length at least k. This bound is tight when k-2 divides n-1. ... The authors would like to thank Jacques Verstraëte*for*suggesting this problem*and**for*sharing his ideas*and*methods used in similar problems. ...##
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Critical hypergraphs and intersecting set-pair systems

1985
*
Journal of combinatorial theory. Series B (Print)
*

An ISP-

doi:10.1016/0095-8956(85)90043-7
fatcat:5q3iozxw25agdcpgzvf4nshblm
*system*is called (a, @-*system*if, additionally, (ii) [Ai/ = a*and*IBil = b*for*every i < m. ... Results are applied*for*T-critical*and*v-critical*hypergraphs*. 0 1985 Academic Press, Inc. (i) Ai"Bj=(2/ if*and*only if i=j whenever 1 <ii, j<m. ... ACKNOWLEDGMENT The author would like to tive*and*helpful comments. express his to A. Gyiirfhs*and*J. Lehel*for*their construc- ...##
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Page 72 of Mathematical Reviews Vol. , Issue 90A
[page]

1990
*
Mathematical Reviews
*

T. (1-CLEM-C);
Laskar, R. (1-CLEM)

*Gallai**theorems**for**graphs*,*hypergraphs*,*and**set**systems*. Proceedings of the First Japan Conference on*Graph*Theory*and*Applications (Hakone, 1986). ... A classical*theorem*of*Gallai*states that ap + Bo = a; + 8; = P. ...##
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The enumeration problem for color critical linear hypergraphs

1980
*
Journal of combinatorial theory. Series B (Print)
*

It is shown that

doi:10.1016/0095-8956(80)90047-7
fatcat:oshqv4oby5dzjo5pn3ydpuw2v4
*for*n 2 3, r > 3 there exists a constant c > 1 depending only on n*and*r such that S(m, n, r) > cm*for*all sufficiently large m. ... Denote by S(m, n, r) the number of non-isomorphic r-critical linear n-*graphs*on m vertices. ... Even in this special case the obtained*hypergraph*X is not necessarily r-critical.*For*2-*graphs**and*r > 4 a counter-example was given by*Gallai*(see Fig. (1) ). ...
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