Habilitation Thesis
Paths, cycles and factors in graphs and hypergraphs
Gyula Katona
Department of Computer Science and Information Theory Budapest University of Technology and Economics
May 2013
Introduction
The history of graph theory started already in the 18th century, but for a long time it was not considered to be a serious science. It started to gain some respect with the appearence of computers in the middle of the 20th century. Many major results that are essential nowdays in computer science where proved in those days. Today graph theory is respected area of both computer science and mathematics.
In the present thesis I summerize my contribution to graph theory in the area of paths, cycles and factors in graphs and hypergraphs.
Paths and cycles are fundamental notions of graphs, their role also shows in many practical application in computer science. Factors of a graphs are generalizations of this notion, it is a well studied area especially in Hungary.
In this thesis I include 3 of my papers in this area.
The corresponding notions in hypergaphs where also studied in the past 30 years, but research in this area was boosted in 1999 when my paper with H. Kiersted appeared [9]. Since then more then 70 pepers where written on this topic. I present 6 of my papers about such results.
1 Paths, cycles and factors in graphs
1.1 (1, f )-odd subgraphs
In [8] we deal with a special case of the degree prescribed subgraph problem, introduced by Lov´asz [43]. This is as follows. Let G be an undirected graph and let∅ 6=Hv ⊆N∪ {0} be a degree prescription for each v ∈V(G). For a spanning subgraphF ofG, define δHF(v) = min{|degF(v)−i|: i∈ Hv}, and let
δFH= X
v∈V(G)
δFH(v) and δH(G) = min
F δHF,
where the minimum is taken over all the spanning subgraphs F of G. A spanning subgraph F is called H-optimal if δFH = δH(G), and it is an H- factor ifδHF = 0, i.e., if degF(v)∈ Hv for allv ∈V(G). The degree prescribed subgraph problem is to determine the value ofδH(G).
An integer h is called a gap of H ⊆ N∪ {0} if h /∈ H but H contains an element less than h and an element greater than h. Lov´asz [45] gave a structural description on the degree prescribed subgraph problem in the case where Hv has no two consecutive gaps for all v ∈ V(G). He showed that the problem is NP-complete without this restriction. The first polynomial time algorithm was given by Cornu´ejols [28]. It is implicit in Cornu´ejols [28]
that this algorithm implies a Gallai–Edmonds type structure theorem for the degree prescribed subgraph problem, which is similar to – but in some respects much more compact than – that of Lov´asz’.
The case when an odd value function f : V(G) → N is given and Hv = {1,3,5, . . . , f(v)} for allv ∈V(G), is called the (1, f)-odd subgraph problem.
We denote δH(G) = δf(G). This problem is much simpler than the general case due to the fact that only parity requirements are posed. The (1, f)-odd subgraph problem was first investigated by Amahashi [23], who gave a Tutte type characterization of graphs having a [1, n]-odd factor, where n≥1 is an odd integer. A Tutte type theorem for a general odd value function f was proved by Cui and Kano [51]. We then gave a Berge type minimax formula
onδf(G) in [6]:
Thesis 1.1 ([6]). The order of a maximum(1, f)-odd subgraph H of a graph G is given by
|H|=|G| − max
S⊆V(G){o(G−S)−X
x∈S
f(x)},
where o(G−S) is the number of odd components of G−S.
We also showed another property of (1, f)-odd subgraphs, which is a generalization of the following property of matchings.
Theorem 1.1 ([32]; [46] p.88). Let G be a graph, and B and R be subsets of V(G) such that |B| <|R|. If there exists a matching which covers B and one which coversR, then there exists a matching which coversB and at least one vertex of R\B.
Thesis 1.2 ([6]). Let G be a graph, and B and R be subsets of V(G) such that |B| <|R|. If there exists a (1, f)-odd subgraph which covers B and one which covers R, then there exists a (1, f)-odd subgraph which covers B and at least one vertex ofR\B. In particular, every maximal (1, f)-odd subgraph is a maximum (1, f)-odd subgraph.
1.2 Structure theorem
In [7] we prove a Gallai–Edmonds type theorem on the (1, f)-odd subgraph problem, first we give the necessary definitions.
For a graph G, define
τ(G) = the order of a maximum (1, f)-odd subgraph ofG.
For any vertex x of G, we denote by Gx the graph obtained from G by adding a new vertex w together with a new edge wx and define f(w) = 1.
Let D(G) denote the set of all vertices x of G such that τ(Gx) = τ(G) + 2.
LetA(G) be the set of vertices ofV(G)−D(G) that are adjacent to at least
one vertex in D(G). Finally, define C(G) = V(G)−D(G)−A(G). Then V(G) is decomposed into three disjoint subsets
V(G) =D(G)∪A(G)∪C(G). (1)
Note that if f(x) = 1 for all vertices x of G, then a maximum (1, f)-odd subgraph is a maximum matching and a vertex y satisfiesτ(Gy) =τ(G) + 2 if and only if y is not contained in a certain maximum matching in G, and thus the above decomposition V(G) = D(G)∪A(G)∪C(G) becomes the Gallai-Edmonds decomposition ([46] p.94).
C(G) D(G)
A(G) v u
Figure 1: DecompositionV(G) =D(G)∪A(G)∪C(G), withf(v) = 1, f(u) = 3 and one uncovered vertex.
Thesis 1.3 ([7]). Let G be a graph, and V(G) = D(G)∪ A(G)∪ C(G) the decomposition defined in (1). Then the following statements hold (see Figure 1):
(i) Every component ofhD(G)iG is critical with respect to(1, f)-odd factor.
(ii) hC(G)iG has a (1, f)-odd factor.
(iii) Every maximum (1, f)-odd subgraph H of G covers C(G)∪A(G), and for every vertex x ∈ A(G), degH(x) = f(x) and every edge of H inci- dent with x joins x to a vertex in D(G).
(iv) The order |H| of a maximum (1, f)-odd subgraph H is given by
|H|=|G|+ω(hD(G)iG)− X
x∈A(G)
f(x), (2)
where ω(hD(G)iG) denotes the number of components of hD(G)iG. In [8] we show a new approach to the (1, f)-odd subgraph problem. Ac- tually, it is worth allowing f to have also even values and defining Hv equal to {1,3, . . . , f(v)} or {0,2, . . . , f(v)}, according to the parity of f(v). We call this the f-parity subgraph problem. We show an easy reduction of the f-parity subgraph problem to the matching problem, and we show that this reduction easily yields the above mentioned Gallai–Edmonds and Berge type theorems on the f-parity subgraph problem.
To show this we use the auxiliary graph defined below:
Definition 1.2 ([8]). For a graph G and a function f : V(G) → N, define Gf to be the following undirected graph. Replace every vertex v ∈V(G) by a new complete graph on f(v) vertices, denoted by Kf(v), and for each pair of vertices u, v ∈ V(G) adjacent in G, add all possible f(u)f(v) edges between Kf(u) and Kf(v). Let Vf(v) =V(Kf(v)).
The constant function f ≡1 is simply denoted by 1. Observe that G1 = G, |V(Gf)| = f(V(G)) and that Vf(v) 6= ∅ for every v ∈ V(G). There is a strong connection between the maximum matchings of Gf and the optimal f-parity subgraphs ofG. Note that the size of a maximum matching ofG is just |V(G)| −δ1(G).
Thesis 1.4 ([8]). For every optimal f-parity subgraph F of G, there exists a matchingM ofGf such that|V(M)|=f(V(G))−δfF. Moreover, ifdegF(w)∈ {. . . , f(w)−3, f(w)−1} for a vertex w ∈ V(G) then M can be chosen to miss a prescribed vertex x ∈ Vf(w). On the other hand, for every maximum matching M of Gf there exists a spanning subgraph F of G such that δfF = f(V(G))− |V(M)|. Moreover, if M misses a vertex in Vf(w) for some w ∈ V(G) then F can be chosen such that degF(w) ∈ {. . . , f(w)−3, f(w)−1}.
In particular, δf(G) =δ1(Gf).
By Thesis 1.4,Gisf-critical if and only ifGf is factor-critical. The Gallai–
Edmonds structure theorem for thef-parity subgraph problem follows from the classical Gallai–Edmonds theorem easily and it is easy to derive a Berge- type formula of Thesis 1.1 as well.
1.3 f -elementary graphs
In [8] we also generalize some results on elementary graphs, obtained in Lov´asz [44], to the f-parity case.
Definition 1.3. Let G be a connected graph and f : V(G) → N. An edge e ∈ E(G) is said to be f-allowed if G has an f-parity factor containing e.
Otherwise e is f-forbidden. The graph G is said to be f-elementary if the f-allowed edges induce a connected spanning subgraph of G. The graph G is weakly f-elementary if G2 is f-elementary, where G2 is the graph obtained from G by replacing every edge e∈E(G) by two parallel edges.
A1-elementary graph is briefly calledelementary. Anf-elementary graph is weakly f-elementary, but not vice versa: G = K2 with f ≡ 2 is weakly f-elementary but not f-elementary. These classes coincide if f =1. Lemma 1.4 justifies why we introduced the weak version of f-elementary graphs.
Lemma 1.4 ([8]). Gf is elementary if and only if Gis weakly f-elementary.
The f = 1 special cases of the following two theorems can be found in Lov´asz and Plummer [46] (Theorems 5.1.3 and 5.1.6). Using our reduction, these special cases imply both Thesis 1.5 and 1.6.
Thesis 1.5 ([8]). A graph G is weakly f-elementary if and only if δf(G) = 0 and Cf−χw(G) = ∅ for all w∈V(G).
Thesis 1.6 ([8]). A graph Gis weaklyf-elementary if and only iff-odd(G− Y) ≤ f(Y) for all Y ⊆ V(G), and if equality holds for some Y 6= ∅ then G−Y has no f-even components.
In the matching case the existence of a certain canonical partition of the vertex set was revealed by Lov´asz [44] (Lov´asz, Plummer [46], Theorem 5.2.2). We cite this result.
Definition 1.5. A set X ⊆V(G) is called nearly f-extreme if δf−χX(G) = δf(G) +|X|. Besides, X is f-extreme if δf(G−X) = δf(G) +f(X).
It is clear that δf−χX(G)≤ δf(G) +|X| and δf(G−X)≤δf(G) +f(X) for every X ⊆V(G). Nearly 1-extreme and 1-extreme sets coincide.
Theorem 1.6 (Lov´asz [44]). If G is elementary then the maximal barriers of G form a partition S of V(G). Moreover, it holds that
1. foru, v ∈V(G), the graphG−u−v has a perfect matching if and only if u andv are contained in different classes of S, (hence an edge xy of G is 1-allowed in G if and only if x and y are contained in different classes of S),
2. S ⊆V(G) is a class of S if and only ifG−S has |S|components, each factor-critical,
3. X ⊆V(G) is1-extreme if and only if X ⊆S for some S ∈ S.
Lemma 1.4 implies the analogue of this result.
Thesis 1.7 ([8]). If G is weakly f-elementary then its maximal f-barriers form a subpartition S0 of V(G). Call the classes of S0 proper, and add all elements v ∈ V(G) not in a class of S0 as a singleton class yielding the partition S of V(G). Now it holds that
1. for u, v ∈ V(G), the graph G has an (f −χ{u,v})-parity factor if and only ifuandv are contained in different classes ofS (hence an edgexy ofG isf-allowed inG2 if and only if xandy are contained in different classes of S),
2. S ⊆ V(G) is a class of S0 if and only if G−S has f(S) components, each f-critical,
3. X ⊆V(G) is nearly f-extreme (f-extreme, resp.) if and only if X ⊆S for someS ∈ S (S ∈ S0, resp.).
Remark It follows from Theorem 1.7 that S could be introduced as the partition {X ⊆V(G) :X is a maximal nearly f-extreme set of G}. Besides, if X ⊆ V(G), |X| ≥ 2 is maximal nearly f-extreme, then X is an f-barrier of G.
Corollary 1.7 ([8]). If G isf-elementary then an edgee isf-allowed if and only if e joins two classes of S.
2 Paths and cycles in hipergraphs
2.1 Hamiltonian cycles
LetH be ak–uniform hypergraph on the vertex set V(H) ={v1, v2, . . . , vn} where n > k. vn+x with x ≥ 0 denotes the same vertex as vx for simplicity of notation. The set of the edges, k–element subsets of V, is denoted by E(H) ={E1, E2, . . . , Em}.
In [9] we gave the following definition:
Definition 2.1. A cyclic ordering (v1, v2, . . . , vn) of the vertex set is called ahamiltonian cycle iff for every1≤i≤n there exists an edgeEj of H such that {vi, vi+1, . . . , vi+k−1}=Ej.
Since an ordinary graph is a 2-uniform hypergraph, this definition gives the definition of the hamiltonian cycle in ordinary graphs for k = 2. (As a matter of fact, in the original paper the termchainwas used instead of cycle, but it seems that everyone prefers cycle.)
The first natural question was to find a Dirac type theorem for hamilto- nian cycles. For this, we need to extend the definition of degree for hyper- graphs.
Definition 2.2. Thedegree of a fixedl–tupleof distinct vertices,{v1, v2, . . . , vl}, in a k–uniform hypergraph is the number of edges of the hypergraph con- taining all {v1, v2, . . . , vl}. It is denoted by d(k)(v1, v2, . . . , vl). Furthermore δ(k)l (H) denotes minimum of d(k)(v1, v2, . . . , vl) over all l-tuples in H. (If k is clear from the context, only δl(H) will be used.)
We conjectured that δ(k)k−1(H) ≥ bn−k+32 c implies the existence of the hamiltonian cycle, and showed that this bound cannot be lowered. From the other side, a Dirac type theorem was proved for any k, however, the degree bound was far from being best possible.
Thesis 2.1 ([9]). If H = (V,E) is a k-uniform hypergraph on n vertices with δk−1(H)>
1− 1
2k
n+ 4−k− 2 k, then H contains a hamiltonian cycle.
For k = 3 the above result requires roughly 56n degree bound for each pair of vertices, but it is conjectured that only 12n is needed.
Now, more then a decade later, the problem is nearly settled. In [47]
Ruci´nski, R¨odl and Szemer´edi proved that the conjecture is asymptotically true for k = 3, then in [49] the exact result was given in this case, together with the analogous hamiltonian path result.
Theorem 2.3([49]). LetH= (V,E)be a3-uniform hypergraph onn vertices where n is sufficiently large. If δ2(H)≥ b12nc thenH contains a hamiltonian cycle. If δ2(H)≥ b12nc −1 then H contains a hamiltonian path.
For larger k values only the asymptotically sharp bound is known.
Theorem 2.4 ([48]). Let k ≥ 3, γ > 0, and let H = be a k-uniform hyper- graph on n vertices where n is sufficiently large. If δk−1(H)≥(12+γ)n, then H contains a hamiltonian cycle.
The proofs of the above results uses similar ideas that appear in the proof of the celebrated Szemer´edi Lemma, but as a result, the proof works only if n is sufficiently large. Although this lower bound on n is not as large as in the original Szemer´edi Lemma, but is is still beyond “normal size”. So the problem is still open for normal size hypergraphs, as well, as the exact degree bound for largerk.
The above definition for hamiltonian cycle is very strict, the consecutive edges in the cycle must intersect ink−1 vertices. It is possible to relax this condition, so one can define loose hamiltonian cycles. Extending the idea, we can define cycles with various tightness.
Definition 2.5. A k-uniform hypergraph is called an l-cycle if there is a cyclic ordering of the vertices such that every edge consists of k consecu- tive vertices, every vertex is contained in an edge and two consecutive edges (where the ordering of the edges is inherited from the ordering of the vertices) intersect in exactly l vertices. Naturally, we say that a k-uniform, n-vertex hypergraphH contains a hamiltonianl-cycle if there is a subhypergraph ofH which forms an l-cycle and which covers all vertices of H.
Note, that to “nicely close” the cycle it is necessary that (k−l)|n. The previous tight hamiltonian cycle is a hamiltonian (k−1)-cycle. (Clearly (k− (k−1))|n always hold.)
The number of edges in a hamiltonian l-cycle is k−ln , and n in a tight hamiltonian cycle. Taking every (k−l)th edge from a tight hamiltonian cycle yields a hamiltonian l-cycle. This shows that in any Dirac-type theorem for hamiltonianl-cycles, the degree bound cannot be more than (12 +o(1))n.
There are several interesting results on these l-cycles [37, 39, 40, 41].
2.2 Extremal questions
The first natural extremal question, which already appeared in [9], asks for the maximum number of edges in a uniform hypergraph containing no hamil- tonian cycle.
For graphs the extremal case is a complete graph on (n −1) vertices completed by a vertex of degree one. It is trivial to prove that this graph does not contain a hamiltonian cycle, but it is not straightforward to prove that this is the extremal case.
For hypergraphs, even determining the extremal case looks more difficult.
An easy double-counting argument gives an upper bound, we count how many hamiltonian cycles are destroyed if an edge is deleted from the complete hypergraph.
Thesis 2.2 ([9]). If H is a k–uniform hypergraph on n vertices satisfying
|E(H)| ≥ n−1 n
n k
then H contains a hamiltonian (k−1)-cycle.
At an earlier stage we found a construction with a fairly large number of edges containing no hamiltonian cycle. Later better constructions were found, but surprisingly this is still the best one in the special case when k = 3 and 3|(n−1).
Thesis 2.3. For any n ≥ 6, if 3|(n−1) then there exists a 3–uniform hy- pergraph H(n,3) which does not contain a hamiltonian cycle and satisfies
|E(H(n,3))|=
n−1 3
+n−1. (3)
This construction is the following. Let us partition the vertex setV −{vn} into q sets V1, . . . , Vq such that |V1| = . . . = |Vq| = 3, since 3|(n −1) this is possible to do. The edge set E(H(n,3)) consists of all 3–element subset of V − {vn}, and all 2–element subset of Vi (1 ≤ i ≤ q) extended with the
vertex vn. It follows immediately that (3) holds, and it is not so difficult to see that this hypergraph does not contain a hamiltonian cycle.
In the general case the best construction is due to Tuza [50]. Using Steiner systems he constructed hypergraphs with no hamiltonian cycles and
|E(H(n, k))|=
n−1 k
+ (1−o(1))
n−2 k−2
.
With Frankl in [3] we improved the upper bound by extending the count- ing argument. We count how many subhypergraphs are there in the complete hypergraph, that contain a hamiltonian cycle, and at least 2 edges need to be removed to destroy all hamiltonian cycles in this subgraph. Determining the minimum number of edges in such hypergraphs leads to the following improvement of the upper bound.
Thesis 2.4 ([3]). If H is a k–uniform hypergraph on n vertices satisfying
|E(H)| ≥ n
r 1− 4k (4k−1)n
then H contains a hamiltonian (k−1)-cycle.
Recently, Glebov, Persons and Weps have settled the 3-uniform case.
Theorem 2.6 ([36]). There exists ann0 such that for any n≥n0 the maxi- mum number of edges in a k-uniform hypergraph containing no hamiltonian (k−1)-cycle is n−13
+n−1 if 3|n−1, and n−13
+n−2 otherwise.
The exact value is still unknown for k≥4.
Meanwhile, the above method raised problems that are interesting on their own.
Definition 2.7. A hypergraph is r-edge-hamiltonian if by the removal of any r edges a hamiltonian hypergraph is obtained. Let fk(n, r) denote the minimum number of edges in an n-vertex k-uniform r-edge-hamiltonian hy- pergraph.
In [3] lower and upper bounds are given for fk(n, r) in various cases.
Thesis 2.5 ([3]). If n≥6 then 14
9 n ≤f3(n,1) ≤ 11
6 n+o(n), 2n ≤f3(n,2) ≤ 13
4 n+o(n), fk(n,1) ≤ 4k−1
2k n+o(n), 3
2n ≤f4(n,1).
In [2] with Dudek and ˙Zak we have some result on a similar extremal question.
Definition 2.8. We say that a hypergraph H is hamiltonian path (cycle) saturated if H does not contain a hamiltonian (k −1)-path (cycle) but by adding any new edge we create a hamiltonian (k−1)-path (cycle) in H. Let pk(n) (k ≥ 2) denote the minimum number of edges in a hamiltonian path saturated k-uniform hypergraph on n vertices, and let ck(n) (k ≥ 2) denote the minimum number of edges in a hamiltonian cycle saturated k-uniform hypergraph on n vertices
Bollob´as [24] posed the problem of finding the minimum number, c2(n), of edges in a hamiltonian cycle saturated graph onn vertices. In 1972 Bondy [25] proved thatc2(n)≥ d3n2 eforn ≥7. Combined results of Clark, Entrigner and Shapiro [27, 26] and Xiaohui, Wenzhou, Chengxue and Yuansheng [42]
show that this bound is sharp apart from a few smaller values of n. The constructions are mostly tricky graphs based on Isaacs’ snarks (see [38]) and generalized Petersen graphs. It was natural to ask the same question for hamiltonian path saturated graphs. Dudek et al. [1] obtained using some modification’s of Isaacs’ snarks that3n−1
2
−2≤p2(n)≤3n−1
2
forn≥54.
The exact value p2(n) = 3n−1
2
for n ≥ 54 was determined by Frick and Singleton [35].
In [2] with proved a general lower bound for pk(n), and upper bound for p3(n).
Thesis 2.6 ([2]).
pk(n)≥
n k
(k(n−k) + 1) = Ω(nk−1).
Thesis 2.7 ([2]). If n≥12, then p3(n)≤ 3√
30
25 n5/2+o(n5/2) =O(n5/2).
2.3 Shorter paths
In [4] we try to generalize the following result of Gallai for hypergraphs.
Theorem 2.9 (Erd˝os-Gallai [33]). LetG be a graph onn vertices containing no path of length k. Then e(G) ≤ 12(k −1)n. Equality holds iff G is the disjoint union of complete graphs on k vertices.
Of course, it would be possible to ask the question forl-cycles of Definition 2.5. We can get quite exact results regarding hypergraphs avoiding (k−1)- tight paths.
Thesis 2.8 ([4]). Let H be an extremal k-uniform hypergraph containing no (k−1)-tight path of length s. Then
(1 +o(1))s−1 k
n k−1
≤ |e(H)| ≤(s−1) n
k−1
However, if we use a different definition, then we can obtain sharper results.
Definition 2.10. Fix k ≥2 and t,1 ≤ t ≤k−1. A Berge path of length s in a hypergraph is a collection of s hyperedges h1, . . . , hs and s+ 1 vertices v1, . . . , vs+1 such that for each 1≤i≤s we have vi, vi+1 ∈hi.
At-tight Berge-path of lengthsin a k-uniform hypergraph is a Berge-path ons+ 1vertices {v1, v2, . . . , vs+1}ands hyperedges{h1, h2, . . . , hs} such that consecutive hyperedges intersect in at least t points.
So one the difference between the two definitions is that the consecutive edges must intersect in exactly t vertices or at least t vertices. The other difference is that in this definition a path must contain a different edge for eachvi, vi+1 along the path, which is not required in the previous definition.
Note that if t = 1, k = 2 then both definitions gives the same as the usual definition for graphs.
Thesis 2.9 ([4]). Fix k ≥ 2 and t,1≤ t ≤ k−1. Fix s large. Let H be an extremal k-uniform hypergraph on nvertices containing no t-tight Berge-path of length s. Then
(1 +o(1))
n t
s
k
s t
≤e(H)≤
n t
s
k
s t
.
Using this result in [36] the authors give an asymptotically tight bound for the maximum number of edges in a hypergraph that doesn’t contain a (k−1)-cycle.
2.4 Related problems
In [3], to prove one of the bounds, the answer to the following question was needed: What is the minimum number of edges in a graph which contains a P4 (a path on 4 distinct vertices) even after the removal of any m of its edges?
This question can be generalized in the following way.
Definition 2.11 (Stability). Let H be a fixed graph. If the graph G has the property that removing any s edges of G, the resulting graph still contains a subgraph isomorphic to H, then we say that G is m-edge-stable. If the graph Ghas the property that removing anys vertices ofG, the resulting graph still contains a subgraph isomorphic toH, then we say that Gis m-vertex-stable.
SE(H;n, m) denotes the minimum number of edges in a m-stable graph on n vertices, and SE(H;m) denotes the minimum number of edges in any m-stable graph (SE(m;H) = minnSE(H;n, m)). Similarly,SV(H;n, m)de-
notes the minimum number of edges in a m-stable graph on n vertices, and SV(H;m) = minnSV(H;n, m).
Problem 2.12. Determine SE(H;m) and SV(H;m) for a given graph H.
Note that for m fixed, SE(H;n, m) and SV(H;n, m) is decreasing inn, as we can just add isolated vertices to getm-stable graphs onn+ 1, n+ 2, . . . vertices. This implies that for any fixed m, SE(H;n, m) = SE(H;m) if n is large enough. (Same holds, for SV(H;m). Also note that SE(H;m) and SV(H;m) is strictly increasing in m.
Thesis 2.10([5]).LetP4 denote the path on 4 distinct vertices. NowSE(P4; 1) = 4, and if m ≥2, then SE(P4;m) = m+lq
2m+ 94 +32m . This proved the conjecture that appeared in [3].
The most interesting unsolved cases areSE(P5;m) andSE(C4;m). This type of question can be also extended to hypergraphs. An interesting problem is to determine SE(C6(3);m).
Later several cases were solved about vertex stability.
Theorem 2.13 ([29, 31]).
SV(Ci;m) = i(m+ 1), for i= 3,4, SV(K4;m) = 5(m+ 1),
SV(Kn;m) = n+m2
, if n is large enough.
In [34] the extremal graphs for SV(Kn;m) were characterized in many cases of n, m.
Theorem 2.14 ([30]). SV(Kn,m; 1) =mn+m+n.
The author’s publications related to the thesis
[1] A. Dudek, G. Y. Katona, and A. Pawe l Wojda. Hamiltonian path satu- rated graphs with small size. Discrete Applied Mathematics, 154(9):1372–
1379, June 2006.
[2] A. Dudek, G. Y. Katona, and A. Zak. Hamilton-chain saturated hyper- graphs. Discrete Mathematics, 310(6-7):1172–1176, 2010.
[3] P. Frankl and G. Y. Katona. Extremal k-edge-hamiltonian hypergraphs.
Discrete Mathematics, 308(8):1415–1424, Apr. 2008.
[4] E. Gy˝ori, G. Y. Katona, and N. Lemons. Hypergraph Extensions of the Erd˝os-Gallai Theorem. Electronic Notes in Discrete Mathematics, 36(78439):655–662, Aug. 2010.
[5] I. Horv´ath and G. Y. Katona. Extremal P4 -stable graphs. Discrete Applied Mathematics, 159(16):1786–1792, 2011.
[6] M. Kano and G. Y. Katona. Odd Subgraphs And Matchings. Discrete Mathematics, 250(1-3):265–272, 2002.
[7] M. Kano and G. Y. Katona. Structure Theorem And Algorithm on (1, f)- odd Subgraph. Discrete Mathematics, 307(11-12):1404–1417, 2007.
[8] M. Kano, G. Y. Katona, and J. Szab´o. Elementary Graphs with Respect tof-Parity Factors.Graphs and Combinatorics, 25(5):717–726, Feb. 2009.
[9] G. Y. Katona and H. A. Kierstead. Hamiltonian Chains in Hypergraphs.
Journal of Graph Theory, 30(3):205–212, 1999.
Further publications of the author
[10] D. Bauer, G. Y. Katona, D. Kratsch, and H. Veldman. Chordality And 2-factors in Tough Graphs. Discrete Applied Mathematics, 99(1-3):323–
329, 2000.
[11] F. G¨oring and G. Y. Katona. Local Topological Toughness and Local Factors. Graphs and Combinatorics, 23(4):387–399, Aug. 2007.
[12] M. Kano, G. Y. Katona, and Z. Kiraly. Packing Paths of Length at Least Two. Discrete Mathematics, 283(1-3):129–135, 2004.
[13] G. Y. Katona. Searching For f-hamiltonian Circuits. Combinatorica, 12(2):241–245, 1992.
[14] G. Y. Katona. Edge Disjoint Polyp Packing. Discrete Applied Mathe- matics, 78(1-3):133–152, Oct. 1997.
[15] G. Y. Katona. Toughness And Edge-toughness. Discrete Mathematics, 164(1-3):187–196, Feb. 1997.
[16] G. Y. Katona. Properties of Edge-tough Graphs. Graphs and Combi- natorics, 15(3):315–325, 1999.
[17] G. Y. Katona. A Large Set of Non-hamiltonian Graphs.Discrete Applied Mathematics, 115(1-3):99–115, Nov. 2001.
[18] G. Y. Katona. Vertex disjoint polyp packing. Annales Univ. Sci. Bu- dapest., Sect. Comp, 21:81–118, 2002.
[19] G. Y. Katona. Hamiltonian chains in hypergraphs, A survey. In S. Aru- mugam, B. D. Acharya, and S. B. Rao, editors,Graphs, Combinatorics, Algorithms and its Applications, pages 71–78, New Delhi CL - Krish- nankoil CC - India, 2005. Narosa Publishing House.
[20] G. Y. Katona, G. O. H. Katona, and Z. Katona. Most Probably Inter- secting Families of Subsets. Combinatorics, Probability and Computing, 21(1-2):219–227, Feb. 2012.
[21] G. Y. Katona and N. Sieben. Bounds on the Rubbling and Optimal Rubbling Numbers of Graphs. Graphs and Combinatorics, Mar. 2012.
[22] L. A. Zahor´anszky, G. Y. Katona, P. H´ari, A. M´aln´asi-Csizmadia, K. A.
Zweig, and G. Zahor´anszky-K¨ohalmi. Breaking the hierarchy–a new cluster selection mechanism for hierarchical clustering methods. Algo- rithms for molecular biology : AMB, 4(1):12, Jan. 2009.
Furdther References
[23] A. Amahashi. On factors with all degrees odd. Graphs Combin., 1(2):111–114, 1985.
[24] B. Bollob´as. Extremal graph theory, volume 11 ofLondon Mathematical Society Monographs. Academic Press Inc. [Harcourt Brace Jovanovich Publishers], London, 1978.
[25] J. A. Bondy. Variations on the Hamiltonian theme. Canad. Math. Bull., 15:57–62, 1972.
[26] L. Clark and R. Entringer. Smallest maximally non-Hamiltonian graphs.
Period. Math. Hungar., 14(1):57–68, 1983.
[27] L. H. Clark, R. C. Entringer, and H. D. Shapiro. Smallest maximally non-Hamiltonian graphs. II. Graphs Combin., 8(3):225–231, 1992.
[28] G. Cornu´ejols. General factors of graphs. J. Combin. Theory Ser. B, 45(2):185–198, 1988.
[29] A. Dudek, A. Szyma´nski, and M. Zwonek. (H,k) stable graphs with minimum size. Discuss. Math. Graph Theory, 28(1):137–149, 2008.
[30] A. Dudek and A. ˙Zak. On vertex stability with regard to complete bipartite subgraphs.Discuss. Math. Graph Theory, 30(4):663–669, 2010.
[31] A. Dudek and M. Zwonek. (H,k) stable bipartite graphs with minimum size. Discuss. Math. Graph Theory, 29(3):573–581, 2009.
[32] J. Edmonds and D. R. Fulkerson. Transversals and matroid partition.
J. Res. Nat. Bur. Standards Sect. B, 69B:147–153, 1965.
[33] P. Erd˝os and T. Gallai. On maximal paths and circuits of graphs. Acta Math. Acad. Sci. Hungar, 10:337—-356 (unbound insert), 1959.
[34] J.-L. Fouquet, H. Thuillier, J.-M. Vanherpe, and A. P. Wojda. On (Kq, k) stable graphs with small k. Electron. J. Combin., 19(2):Paper 50, 10, 2012.
[35] M. Frick and J. Singleton. Lower bound for the size of maximal non- traceable graphs. Electron. J. Combin., 12:Research Paper 32, 9 pp.
(electronic), 2005.
[36] R. Glebov, Y. Person, and W. Weps. On extremal hypergraphs for Hamiltonian cycles. European J. Combin., 33(4):544–555, 2012.
[37] H. H´an and M. Schacht. Dirac-type results for loose Hamilton cycles in uniform hypergraphs. J. Combin. Theory Ser. B, 100(3):332–346, 2010.
[38] R. Isaacs. Infinite families of nontrivial trivalent graphs which are not Tait colorable. Amer. Math. Monthly, 82:221–239, 1975.
[39] P. Keevash, D. K¨uhn, R. Mycroft, and D. Osthus. Loose Hamilton cycles in hypergraphs. Discrete Math., 311(7):544–559, 2011.
[40] D. K¨uhn, R. Mycroft, and D. Osthus. Hamilton `-cycles in uniform hypergraphs. J. Combin. Theory Ser. A, 117(7):910–927, 2010.
[41] D. K¨uhn and D. Osthus. Loose Hamilton cycles in 3-uniform hyper- graphs of high minimum degree. J. Combin. Theory Ser. B, 96(6):767–
821, 2006.
[42] X. Lin, W. Jiang, C. Zhang, and Y. Yang. On smallest maximally non- Hamiltonian graphs. Ars Combin., 45:263–270, 1997.
[43] L. Lov´asz. The factorization of graphs. In Combinatorial Structures and their Applications (Proc. Calgary Internat. Conf., Calgary, Alta., 1969), pages 243–246. Gordon and Breach, New York, 1970.
[44] L. Lov´asz. On the structure of factorizable graphs. I, II. Acta Math.
Acad. Sci. Hungar., 23:179—-195; ibid. 23 (1972), 465—-478, 1972.
[45] L. Lov´asz. The factorization of graphs. II. Acta Math. Acad. Sci. Hun- gar., 23:223–246, 1972.
[46] L. Lov´asz and M. D. Plummer. Matching theory. AMS Chelsea Pub- lishing, Providence, RI, 2009.
[47] V. R¨odl, A. Ruci´nski, and E. Szemer´edi. A Dirac-type theorem for 3- uniform hypergraphs. Combin. Probab. Comput., 15(1-2):229–251, 2006.
[48] V. R¨odl, A. Ruci´nski, and E. Szemer´edi. An approximate Dirac-type the- orem for k-uniform hypergraphs. Combinatorica, 28(2):229–260, 2008.
[49] V. R¨odl, A. Ruci´nski, and E. Szemer´edi. Dirac-type conditions for Hamiltonian paths and cycles in 3-uniform hypergraphs. Adv. Math., 227(3):1225–1299, 2011.
[50] Z. Tuza. Steiner systems and large non-Hamiltonian hypergraphs.
Matematiche (Catania), 61(1):179–183, 2006.
[51] C. Yuting and M. Kano. Some results on odd factors of graphs.J. Graph Theory, 12(3):327–333, 1988.
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