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### Gallai theorems for graphs, hypergraphs, and set systems

E.J. Cockayne, S.T. Hedetniemi, R. Laskar
1988 Discrete Mathematics
Acknowledgement The authors thank the referee 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.  ...

### Connected τ -critical hypergraphs of minimal size

Matěj Stehlík
2005 Discrete Mathematics & Theoretical Computer Science
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  ...  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.  ...

### Extensions of Gallai's graph covering theorems for uniform hypergraphs

Zsolt Tuza
1991 Journal of combinatorial theory. Series B (Print)
Thus, rk" 6 rf" + (r -1) f', r(k" -f") < (r -1)f' < (r -l)p, 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  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.  ...

### Motivations and history of some of my conjectures

Claude Berge
1997 Discrete Mathematics
If H is the hypergraph dual of a Steiner triple system, the same bound for q(H) was also conjectured by J. Colbourn and M. Colbourn .  ...  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  ...

### Minimum number of elements representing a set system of given rank

Zsolt Tuza
1989 Journal of combinatorial theory. Series A
Answering a 25-year-old problem of Erdos 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.  ...

### Coloring hypergraphs of low connectivity [article]

Thomas Schweser, Michael Stiebitz, Bjarne Toft
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)]).  ...

### 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

Gyula Y. Katona
2014 Electronic Notes in Discrete Mathematics
There are many results 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  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  ) 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.  ...

### Color-Critical Graphs and Hypergraphs with Few Edges: A Survey [chapter]

Alexandr Kostochka
2006 Bolyai Society Mathematical Studies
The current situation with bounds on the smallest number of edges in colorcritical 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 .  ...

### 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.  ...

### Color-critical Graphs and Hereditary Hypergraphs [article]

András Sebő
2019 arXiv   pre-print
A quick proof of Gallai's celebrated 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)?  ...

### Avoiding long Berge cycles [article]

Zoltan Furedi, Alexandr Kostochka, Ruth Luo
2018 arXiv   pre-print
We also show that the bound is attained only 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.  ...

### Critical hypergraphs and intersecting set-pair systems

Zsolt Tuza
1985 Journal of combinatorial theory. Series B (Print)
An ISP-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-  ...

### 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.  ...

### The enumeration problem for color critical linear hypergraphs

H.L Abbott, A Liu, B Toft
1980 Journal of combinatorial theory. Series B (Print)
It is shown that 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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