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A Median-Type Condition for Graph Tiling
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Electronic Notes in Discrete Mathematics 61 (2017) 979–985 www.elsevier.com/locate/endm
A Median-Type Condition for Graph Tiling Diana Piguet 1,2 Maria Saumell 1,2 Institute of Computer Science The Czech Academy of Sciences Prague, Czech Republic
Abstract Koml´os [Koml´os: Tiling Tur´ an Theorems, Combinatorica, 2000] determined the asymptotically optimal minimum degree condition for covering a given proportion of vertices of a host graph by vertex-disjoint copies of a fixed graph H. We show that the minimum degree condition can be relaxed in the sense that we require only a given fraction of vertices to have the prescribed degree. Keywords: Extremal graph theory, graph tiling, regularity lemma, LP-duality.
1
Introduction and results
A classical question in extremal graph theory is to inquire which density condition of a host graph guarantees the containment of a certain subgraph H. For example the fundamental Tur´an theorem [15] gives the minimal value of the average degree that forces a clique. Similarly, the Erd˝os–Stone theorem [5] 1
Piguet and Saumell were supported by the Czech Science Foundation, grant number GJ16-07822Y. Saumell was also supported by Project LO1506 of the Czech Ministry of Education, Youth and Sports. With institutional support RVO:67985807. 2 Email: {piguet, saumell}@cs.cas.cz http://dx.doi.org/10.1016/j.endm.2017.07.062 1571-0653/© 2017 Elsevier B.V. All rights reserved.
980
D. Piguet, M. Saumell / Electronic Notes in Discrete Mathematics 61 (2017) 979–985
essentially determines the average degree condition guaranteeing the containment of a fixed non-bipartite graph H. Generalising this approach of seeking for a single copy of a graph H, researchers have been investigating conditions guaranteeing an H-tiling of size m in a graph G, i.e., the containment of m |V (G)| vertex-disjoint copies of H. The tiling is perfect if m = |V . (H)| The Hajnal-Szemer´edi theorem on equitable colouring [7] determines the minimum degree condition guaranteeing a perfect clique tiling. It is not hard to deduce from their result the minimum degree condition needed to force a partial clique-tiling of any given size. Regarding the average degree condition guaranteeing a clique-tiling, not much has been done, yet. Erd˝os-Gallai [4] determined the required average degree that forces a matching of a given size and Allen et al. [1] the one forcing a triangle tiling. Determining an optimal average degree condition for clique-tiling of higher order is wide open and no conjecture has even been formulated, yet. Let us now move to tilings with an arbitrary graph H. Extending the Hajnal-Szemer´edi theorem, Koml´os [10] obtained a minimum degree bound that forces an H-tiling of a given size. His result states that if H is an r1 colourable graph with colour class sizes 1 ≥ . . . ≥ r > 0, x ∈ (0, |V (H)| ) and n G is an n-vertex graph with minimum degree at least (r − 2 + xr ) r−1 , then G contains an H-tiling of size at least (x − o(1))n. Note that H may have many r-colourings. Of course, in order to apply Koml´os’ theorem with the weakest hypothesis, we need to fix an r-coloring of H which minimizes the size of its smallest colour class. For the corresponding value of r and for each x, Koml´os constructs graphs, which he calls bottleneck graphs and which show asymptotic optimality of his result. Koml´os’ tiling theorem was complemented by K¨ uhn and Osthus [11] who established the optimal minimum degree condition forcing a perfect H-tiling. Considering average degree rather than minimum degree, Grosu and Hladk´ y [6] extended Erd˝os–Gallai theorem, by asymptotically establishing the average degree condition forcing the containment of an H-tiling, for a fixed bipartite graph H. A finer approach to extremal problems is to take into account more information encoded in the degree sequence of the host graph. Let us give an example in the area of tree containment. It is trivial to see that a minimum degree of k ensures a copy of any tree of size k. However, only half of the vertices might need to have this degree to guarantee the same assertion, as conjectured by Loebl, Koml´os, and S´os. Taking this perspective in the area of tilings, Treglown asymptotically determined an optimal degree sequence condition forcing a perfect H-tiling [14]. In this paper, we inquire what portion of vertices need to meet the degree bound in Koml´os’ tiling theorem in order
D. Piguet, M. Saumell / Electronic Notes in Discrete Mathematics 61 (2017) 979–985
981
to guarantee an H-tiling of a given size. Our main result is the following. Theorem 1.1 Let H be an r-colourable graph with colour class sizes 1 ≥ 1 . . . ≥ r > 0 and let x ∈ (0, |V (H)| ). Then for any ε > 0 there is an n0 ∈ N n the following holds. such that for any n ≥ n0 and for δ := (r − 2 + xr ) r−1 n vertices of Any n-vertex graph with at least (1 + ε)(r − 2 + x|V (H)|) r−1 degree at least (1 + ε)δ contains an H-tiling of size at least xn.
V1
V2
V3 L
S
Fig. 1. The extremal graph G(x, |V (H)|, r, r , n).
Clearly, Theorem 1.1 strengthens Koml´os’ tiling theorem: the degree bound δ is the same but we do not require all the vertices of the host graph to meet (H)| this degree bound, but rather only roughly a f = r−2+x|V fraction of them. r−1 1 r−2 Note that as x ranges from 0 to |V (H)| , f ranges from r−1 to 1. We shall show that when r is chosen according to the colouring minimizing the size of the smallest colour class, Theorem 1.1 is asymptotically optimal for each value of x. To this end, we construct an n-vertex graph G = G(x, |V (H)|, r, r , n) n n with (r − 2 + x|V (H)|) r−1 vertices of degree δ := (r − 2 + xr ) r−1 and no H-tiling of size more than xn. The vertex set of G is partitioned into four n n sets, |V1 | = xr n, |V2 | = x(|V (H)| − r ) r−1 , |V3 | = (r − 2)(1 − xr ) r−1 , and n |S| = (1 − x|V (H)|) r−1 . The sets V1 , V2 and S are independent, and the
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set V3 induces a balanced complete (r − 2)-partite graph. There is a complete bipartite graph between V3 and V (G) \ V3 and a complete bipartite graph between V1 and V2 , and no other edges. The graph G(x, |V (H)|, r, r , n) is depicted in Figure 1. Note that the vertices in the set L = V1 ∪ V2 ∪ V3 meet n our degree assumption and we have |L| = (r − 2 + x|V (H)|) r−1 . We claim that each copy of H in G must have at least r vertices in V1 . Suppose it is not the case. We can then recolour H as follows so that the smallest colour class has less than r vertices: give the vertices embedded in V1 colour 1, the ones embedded in V2 ∪ S colour 2, and distribute the r − 2 other colours to vertices embedded in V3 . In this colouring, the smallest colour class has at most as many vertices as colour 1 has, which is less than r . Since |V1 | = xr n, we conclude that there cannot be a H-tiling of size more than xn. The proof of Theorem 1.1 relies on the regularity method and on LPduality. The use of LP-duality for clique-tiling has been used in [12]. However, we think that this method deserves more attention and is exposed here in the more general form for H-tiling. Similar use of LP-duality for graph-tiling was used in [3,8,9] but in the context of graphons.
2
Sketch of the proof of Theorem 1.1
Let H be a fixed r-colourable graph and let G be a graph satisfying the degree condition of Theorem 1.1. We apply the regularity lemma [13] on G and erase all edges within clusters, in irregular pairs, and in regular pairs of n low density. Now, at least (1 + 2ε )(r − 2 + x|V (H)|) r−1 vertices have degree at ε least (1 + 2 )δ. Slightly abusing notation, we still call this subgraph G. Let G be the corresponding cluster graph. The Embedding Lemma [2, Lemma 7.5.2] implies that a copy of H in G gives a copy of H in the original graph G; actually it generates roughly |V (G)|/|V (G)| many disjoint copies of H. But not only copies of H in G may generate copies of H in G. For example, if there is a triangle in G, we know there are many copies of C5 in G. Therefore instead of looking for copies of H in G, we shall seek copies of H , where H is some homomorphic image of H. This leads us to the following definition. Let H be a fixed graph. Set HH := {H : there is a homomorphism hH : H → H } . A function f : {H ⊆ G : H ∈ HH } → [0, 1] is a fractional homH -tiling of size k ∈ R+ , if it satisfies the following two properties. (i) For any vertex v ∈ V (G), we have f (Hv ) · |h−1 H (v)| ≤ 1, where the sum
D. Piguet, M. Saumell / Electronic Notes in Discrete Mathematics 61 (2017) 979–985
983
runs over all subgraphs H ⊆ G such that H ∈ HH and v ∈ V (H ). (ii) We have that f (H ) = k, where the sum runs through all subgraphs H ⊆ G with H ∈ HH . The fractional homH -tiling number of a graph G is the maximum of the sizes of all its fractional homH -tilings. In order to prove Theorem 1.1, we shall use LP-duality. The dual notion of fractional homH -tiling is the following. A function c : V (G) → [0, 1] is a fractional homH -cover of size k ∈ R+ , if it satisfies the following two properties. c(v)|h−1 (i) For any subgraph H ⊆ G such that H ∈ HH , we have H (v)| ≥ 1, where the sum runs through all vertices v ∈ V (H ). (ii) We have c(v) = k, where the sum runs through all vertices v ∈ V (G). The fractional homH -cover number of a graph G is the minimum of the sizes of all its fractional homH -covers. The following statement is a straightforward generalisation of the LPduality between the fractional matching and the fractional vertex-cover. Proposition 2.1 For any graph G, its fractional homH -tiling number equals its fractional homH -cover number. Using the regularity method, in order to show that the original graph G has an integral tiling number of at least xn, it is enough to show that the corresponding cluster graph G has fractional tiling number at least (1 + 2ε )x|V (G)|. From Proposition 2.1 we infer that we need to show the following. Proposition 2.2 Let H be an r-colourable graph with colour class sizes 1 ≥ 1 n . . . ≥ r > 0 and let x ∈ (0, |V (H)| ). Set δ := (r − 2 + xr ) r−1 . n Then any n-vertex graph G with at least (r − 2 + |V (H)|) r−1 vertices of degree at least δ has fractional homH -cover number least xn. Proof (Sketch) Let c be a fractional homH -cover of G, and let c denote its size. We need to show that ||c|| = v∈V (G) c(v) ≥ xn. Let L denote the set of vertices of G having degree at least δ, and let S denote the set containing the remaining vertices of G. For any vertex v of G, we denote the set of neighbours of v in L and S by NL (v) and NS (v), respectively. Let v1 be a vertex in L with smallest c(v1 ). Among all the elements in NL (v1 ), let v2 be a vertex with smallest c(v2 ). We continue selecting vertices v3 , v4 , . . . , vr−1 in this way. More precisely, suppose that we have already
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selected i < r − 1 vertices v1 , v2 , . . . , vi in L. We define: NL (i) =
i j=1
NL (vj ),
NS (i) =
i
NS (vj ) .
j=1
Among all the elements in NL (i), we select a vertex vi+1 with smallest c(vi+1 ). After v1 , v2 , . . . , vr−1 have been selected, let u ∈ NL (r −1) be a vertex with the smallest c(u) and w ∈ NS (r −1) one with smallest c(w). It may happen that u or w does not exist, but at least one of them does. For i = 1, 2, . . . , r − 1, we define αi = c(vi ). Set αL = c(u) if u exists, and αL = 1 otherwise. Similarly, we define αS = c(w), or αS = 1. Let αr = min{αL , αS }. The rest of the proof consists of bounding c from below:
c ≥ |L| · α1 +
r−2
|NL (i)| · (αi+1 − αi ) + |NL (r − 1)| · (αL − αr−1 )
i=1
+ |NS (r − 1)| · αS .
(1)
Since c is a fractional homH -cover, we have i i αi ≥ 1. Since all the vertices v1 , . . . , vr−1 are in L, we can provide a lower bound for the size of the common n intersection of the sets NL (vj )∪NS (vj ), |NL (i)|+|NS (i)| ≥ (r−1−i+ixr ) r−1 . Next, it follows from the choice of the vertices vi that the terms αi+1 −αi above are nonnegative. Last, calculations which require a technical case distinction depending on the values of αi give that |NL (r −1)|(αr −αr−1 )+|NS (r −1)|·αr ≥ αr ·r ·xn−αr−1 ((r −1)δ −(r −2)n)) . Plugging the above observations into (1) we can derive that c ≥ xn which finishes the proof. 2
References [1] Allen, P., J. B¨ ottcher, J. Hladk´ y and D. Piguet, A density Corr´ adi-Hajnal theorem, Canad. J. Math. 67 (2015), pp. 721–758. [2] Diestel, R., “Graph theory”, Graduate Texts in Mathematics 173, SpringerVerlag, Berlin, 2005, third edition, xvi+411 pp. [3] Doleˇzal, M. and J. Hladk´ y, Matching polytons, arXiv:1606.06958.
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[4] Erd˝os, P. and T. Gallai, On maximal paths and circuits of graphs, Acta Math. Acad. Sci. Hungar 10 (1959), pp. 337–356 (unbound insert). [5] Erd˝os, P. and A. H. Stone, On the structure of linear graphs, Bulletin of the American Mathematical Society 52 (1946), pp. 1087–1091. [6] Grosu, C. and J. Hladk´ y, The extremal function for partial bipartite tilings, European J. Combin. 33 (2012), pp. 807–815. [7] Hajnal, A. and E. Szemer´edi, Proof of a conjecture of P. Erd˝ os, in: Combinatorial theory and its applications, II (Proc. Colloq., Balatonf¨ ured, 1969), North-Holland, Amsterdam, 1970 pp. 601–623. [8] Hladk´ y, J., P. Hu and D. Piguet, Koml´ os’s tiling theorem via graphon covers, preprint. [9] Hladk´ y, J., P. Hu and D. Piguet, Tilings in graphons, arXiv:1606.03113. [10] Koml´os, J., Tiling Tur´ an theorems, Combinatorica 20 (2000), pp. 203–218. [11] K¨ uhn, D. and D. Osthus, The minimum degree threshold for perfect graph packings, Combinatorica 29 (2009), pp. 65–107. [12] Martin, R. R. and J. Skokan, Asymptotic multipartite version of the Alon-Yuster theorem, J. Comb. Th. B., accepted, arXiv:1307.5897. [13] Szemer´edi, E., Regular partitions of graphs, in: Probl`emes combinatoires et th´eorie des graphes (Colloq. Internat. CNRS, Univ. Orsay, Orsay, 1976), Colloq. Internat. CNRS 260, CNRS, Paris, 1978 pp. 399–401. [14] Treglown, A., A degree sequence Hajnal–Szemer´edi theorem, J. Combin. Theory Ser. B 118 (2016), pp. 13–43. [15] Tur´an, P., On an extremal problem in graph theory (in Hungarian), Mat. Fiz. Lapok 48 (1941), pp. 436–452.
Electronic Notes in Discrete Mathematics 61 (2017) 979–985 www.elsevier.com/locate/endm
A Median-Type Condition for Graph Tiling Diana Piguet 1,2 Maria Saumell 1,2 Institute of Computer Science The Czech Academy of Sciences Prague, Czech Republic
Abstract Koml´os [Koml´os: Tiling Tur´ an Theorems, Combinatorica, 2000] determined the asymptotically optimal minimum degree condition for covering a given proportion of vertices of a host graph by vertex-disjoint copies of a fixed graph H. We show that the minimum degree condition can be relaxed in the sense that we require only a given fraction of vertices to have the prescribed degree. Keywords: Extremal graph theory, graph tiling, regularity lemma, LP-duality.
1
Introduction and results
A classical question in extremal graph theory is to inquire which density condition of a host graph guarantees the containment of a certain subgraph H. For example the fundamental Tur´an theorem [15] gives the minimal value of the average degree that forces a clique. Similarly, the Erd˝os–Stone theorem [5] 1
Piguet and Saumell were supported by the Czech Science Foundation, grant number GJ16-07822Y. Saumell was also supported by Project LO1506 of the Czech Ministry of Education, Youth and Sports. With institutional support RVO:67985807. 2 Email: {piguet, saumell}@cs.cas.cz http://dx.doi.org/10.1016/j.endm.2017.07.062 1571-0653/© 2017 Elsevier B.V. All rights reserved.
980
D. Piguet, M. Saumell / Electronic Notes in Discrete Mathematics 61 (2017) 979–985
essentially determines the average degree condition guaranteeing the containment of a fixed non-bipartite graph H. Generalising this approach of seeking for a single copy of a graph H, researchers have been investigating conditions guaranteeing an H-tiling of size m in a graph G, i.e., the containment of m |V (G)| vertex-disjoint copies of H. The tiling is perfect if m = |V . (H)| The Hajnal-Szemer´edi theorem on equitable colouring [7] determines the minimum degree condition guaranteeing a perfect clique tiling. It is not hard to deduce from their result the minimum degree condition needed to force a partial clique-tiling of any given size. Regarding the average degree condition guaranteeing a clique-tiling, not much has been done, yet. Erd˝os-Gallai [4] determined the required average degree that forces a matching of a given size and Allen et al. [1] the one forcing a triangle tiling. Determining an optimal average degree condition for clique-tiling of higher order is wide open and no conjecture has even been formulated, yet. Let us now move to tilings with an arbitrary graph H. Extending the Hajnal-Szemer´edi theorem, Koml´os [10] obtained a minimum degree bound that forces an H-tiling of a given size. His result states that if H is an r1 colourable graph with colour class sizes 1 ≥ . . . ≥ r > 0, x ∈ (0, |V (H)| ) and n G is an n-vertex graph with minimum degree at least (r − 2 + xr ) r−1 , then G contains an H-tiling of size at least (x − o(1))n. Note that H may have many r-colourings. Of course, in order to apply Koml´os’ theorem with the weakest hypothesis, we need to fix an r-coloring of H which minimizes the size of its smallest colour class. For the corresponding value of r and for each x, Koml´os constructs graphs, which he calls bottleneck graphs and which show asymptotic optimality of his result. Koml´os’ tiling theorem was complemented by K¨ uhn and Osthus [11] who established the optimal minimum degree condition forcing a perfect H-tiling. Considering average degree rather than minimum degree, Grosu and Hladk´ y [6] extended Erd˝os–Gallai theorem, by asymptotically establishing the average degree condition forcing the containment of an H-tiling, for a fixed bipartite graph H. A finer approach to extremal problems is to take into account more information encoded in the degree sequence of the host graph. Let us give an example in the area of tree containment. It is trivial to see that a minimum degree of k ensures a copy of any tree of size k. However, only half of the vertices might need to have this degree to guarantee the same assertion, as conjectured by Loebl, Koml´os, and S´os. Taking this perspective in the area of tilings, Treglown asymptotically determined an optimal degree sequence condition forcing a perfect H-tiling [14]. In this paper, we inquire what portion of vertices need to meet the degree bound in Koml´os’ tiling theorem in order
D. Piguet, M. Saumell / Electronic Notes in Discrete Mathematics 61 (2017) 979–985
981
to guarantee an H-tiling of a given size. Our main result is the following. Theorem 1.1 Let H be an r-colourable graph with colour class sizes 1 ≥ 1 . . . ≥ r > 0 and let x ∈ (0, |V (H)| ). Then for any ε > 0 there is an n0 ∈ N n the following holds. such that for any n ≥ n0 and for δ := (r − 2 + xr ) r−1 n vertices of Any n-vertex graph with at least (1 + ε)(r − 2 + x|V (H)|) r−1 degree at least (1 + ε)δ contains an H-tiling of size at least xn.
V1
V2
V3 L
S
Fig. 1. The extremal graph G(x, |V (H)|, r, r , n).
Clearly, Theorem 1.1 strengthens Koml´os’ tiling theorem: the degree bound δ is the same but we do not require all the vertices of the host graph to meet (H)| this degree bound, but rather only roughly a f = r−2+x|V fraction of them. r−1 1 r−2 Note that as x ranges from 0 to |V (H)| , f ranges from r−1 to 1. We shall show that when r is chosen according to the colouring minimizing the size of the smallest colour class, Theorem 1.1 is asymptotically optimal for each value of x. To this end, we construct an n-vertex graph G = G(x, |V (H)|, r, r , n) n n with (r − 2 + x|V (H)|) r−1 vertices of degree δ := (r − 2 + xr ) r−1 and no H-tiling of size more than xn. The vertex set of G is partitioned into four n n sets, |V1 | = xr n, |V2 | = x(|V (H)| − r ) r−1 , |V3 | = (r − 2)(1 − xr ) r−1 , and n |S| = (1 − x|V (H)|) r−1 . The sets V1 , V2 and S are independent, and the
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set V3 induces a balanced complete (r − 2)-partite graph. There is a complete bipartite graph between V3 and V (G) \ V3 and a complete bipartite graph between V1 and V2 , and no other edges. The graph G(x, |V (H)|, r, r , n) is depicted in Figure 1. Note that the vertices in the set L = V1 ∪ V2 ∪ V3 meet n our degree assumption and we have |L| = (r − 2 + x|V (H)|) r−1 . We claim that each copy of H in G must have at least r vertices in V1 . Suppose it is not the case. We can then recolour H as follows so that the smallest colour class has less than r vertices: give the vertices embedded in V1 colour 1, the ones embedded in V2 ∪ S colour 2, and distribute the r − 2 other colours to vertices embedded in V3 . In this colouring, the smallest colour class has at most as many vertices as colour 1 has, which is less than r . Since |V1 | = xr n, we conclude that there cannot be a H-tiling of size more than xn. The proof of Theorem 1.1 relies on the regularity method and on LPduality. The use of LP-duality for clique-tiling has been used in [12]. However, we think that this method deserves more attention and is exposed here in the more general form for H-tiling. Similar use of LP-duality for graph-tiling was used in [3,8,9] but in the context of graphons.
2
Sketch of the proof of Theorem 1.1
Let H be a fixed r-colourable graph and let G be a graph satisfying the degree condition of Theorem 1.1. We apply the regularity lemma [13] on G and erase all edges within clusters, in irregular pairs, and in regular pairs of n low density. Now, at least (1 + 2ε )(r − 2 + x|V (H)|) r−1 vertices have degree at ε least (1 + 2 )δ. Slightly abusing notation, we still call this subgraph G. Let G be the corresponding cluster graph. The Embedding Lemma [2, Lemma 7.5.2] implies that a copy of H in G gives a copy of H in the original graph G; actually it generates roughly |V (G)|/|V (G)| many disjoint copies of H. But not only copies of H in G may generate copies of H in G. For example, if there is a triangle in G, we know there are many copies of C5 in G. Therefore instead of looking for copies of H in G, we shall seek copies of H , where H is some homomorphic image of H. This leads us to the following definition. Let H be a fixed graph. Set HH := {H : there is a homomorphism hH : H → H } . A function f : {H ⊆ G : H ∈ HH } → [0, 1] is a fractional homH -tiling of size k ∈ R+ , if it satisfies the following two properties. (i) For any vertex v ∈ V (G), we have f (Hv ) · |h−1 H (v)| ≤ 1, where the sum
D. Piguet, M. Saumell / Electronic Notes in Discrete Mathematics 61 (2017) 979–985
983
runs over all subgraphs H ⊆ G such that H ∈ HH and v ∈ V (H ). (ii) We have that f (H ) = k, where the sum runs through all subgraphs H ⊆ G with H ∈ HH . The fractional homH -tiling number of a graph G is the maximum of the sizes of all its fractional homH -tilings. In order to prove Theorem 1.1, we shall use LP-duality. The dual notion of fractional homH -tiling is the following. A function c : V (G) → [0, 1] is a fractional homH -cover of size k ∈ R+ , if it satisfies the following two properties. c(v)|h−1 (i) For any subgraph H ⊆ G such that H ∈ HH , we have H (v)| ≥ 1, where the sum runs through all vertices v ∈ V (H ). (ii) We have c(v) = k, where the sum runs through all vertices v ∈ V (G). The fractional homH -cover number of a graph G is the minimum of the sizes of all its fractional homH -covers. The following statement is a straightforward generalisation of the LPduality between the fractional matching and the fractional vertex-cover. Proposition 2.1 For any graph G, its fractional homH -tiling number equals its fractional homH -cover number. Using the regularity method, in order to show that the original graph G has an integral tiling number of at least xn, it is enough to show that the corresponding cluster graph G has fractional tiling number at least (1 + 2ε )x|V (G)|. From Proposition 2.1 we infer that we need to show the following. Proposition 2.2 Let H be an r-colourable graph with colour class sizes 1 ≥ 1 n . . . ≥ r > 0 and let x ∈ (0, |V (H)| ). Set δ := (r − 2 + xr ) r−1 . n Then any n-vertex graph G with at least (r − 2 + |V (H)|) r−1 vertices of degree at least δ has fractional homH -cover number least xn. Proof (Sketch) Let c be a fractional homH -cover of G, and let c denote its size. We need to show that ||c|| = v∈V (G) c(v) ≥ xn. Let L denote the set of vertices of G having degree at least δ, and let S denote the set containing the remaining vertices of G. For any vertex v of G, we denote the set of neighbours of v in L and S by NL (v) and NS (v), respectively. Let v1 be a vertex in L with smallest c(v1 ). Among all the elements in NL (v1 ), let v2 be a vertex with smallest c(v2 ). We continue selecting vertices v3 , v4 , . . . , vr−1 in this way. More precisely, suppose that we have already
984
D. Piguet, M. Saumell / Electronic Notes in Discrete Mathematics 61 (2017) 979–985
selected i < r − 1 vertices v1 , v2 , . . . , vi in L. We define: NL (i) =
i j=1
NL (vj ),
NS (i) =
i
NS (vj ) .
j=1
Among all the elements in NL (i), we select a vertex vi+1 with smallest c(vi+1 ). After v1 , v2 , . . . , vr−1 have been selected, let u ∈ NL (r −1) be a vertex with the smallest c(u) and w ∈ NS (r −1) one with smallest c(w). It may happen that u or w does not exist, but at least one of them does. For i = 1, 2, . . . , r − 1, we define αi = c(vi ). Set αL = c(u) if u exists, and αL = 1 otherwise. Similarly, we define αS = c(w), or αS = 1. Let αr = min{αL , αS }. The rest of the proof consists of bounding c from below:
c ≥ |L| · α1 +
r−2
|NL (i)| · (αi+1 − αi ) + |NL (r − 1)| · (αL − αr−1 )
i=1
+ |NS (r − 1)| · αS .
(1)
Since c is a fractional homH -cover, we have i i αi ≥ 1. Since all the vertices v1 , . . . , vr−1 are in L, we can provide a lower bound for the size of the common n intersection of the sets NL (vj )∪NS (vj ), |NL (i)|+|NS (i)| ≥ (r−1−i+ixr ) r−1 . Next, it follows from the choice of the vertices vi that the terms αi+1 −αi above are nonnegative. Last, calculations which require a technical case distinction depending on the values of αi give that |NL (r −1)|(αr −αr−1 )+|NS (r −1)|·αr ≥ αr ·r ·xn−αr−1 ((r −1)δ −(r −2)n)) . Plugging the above observations into (1) we can derive that c ≥ xn which finishes the proof. 2
References [1] Allen, P., J. B¨ ottcher, J. Hladk´ y and D. Piguet, A density Corr´ adi-Hajnal theorem, Canad. J. Math. 67 (2015), pp. 721–758. [2] Diestel, R., “Graph theory”, Graduate Texts in Mathematics 173, SpringerVerlag, Berlin, 2005, third edition, xvi+411 pp. [3] Doleˇzal, M. and J. Hladk´ y, Matching polytons, arXiv:1606.06958.
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