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Proof of the Alon–Yuster conjecture

János Komlós, Gábor Sárközy, Endre Szemerédi
2001 Discrete Mathematics  
In this paper we prove the following conjecture of Alon and Yuster. Let H be a graph with h vertices and chromatic number k.  ...  In fact, we show that if H has a k-coloring with color-class sizes h16h26 · · · 6h k , then the conjecture is true with c(H )=h k +h k−1 −1.  ...  For a general H , Alon and Yuster [1] showed that if (G)¿(1 − 1=k)hn, then G contains (1 − O(1))n vertex disjoint copies of H (an 'almost H -factor').  ... 
doi:10.1016/s0012-365x(00)00279-x fatcat:omiqo3umubbajdkuee33r74py4

$K_{\ell}^{-}$-factors in graphs

Daniela Kühn, Deryk Osthus
2005 Discrete Mathematics & Theoretical Computer Science  
This is best possible up to the error term $γn$ and yields an approximate solution to a conjecture of Kawarabayashi.  ...  disjoint copies of $K_ℓ^-$ which covers all vertices of $G$.  ...  This conjecture was proved by Shokoufandeh and Zhao [SZ03] . As in the above conjecture of Alon and Yuster, there are graphs H for which the error term cannot be omitted completely.  ... 
doi:10.46298/dmtcs.3403 fatcat:7hfdtpeitrgl5o2mptkxrklqgm

Page 1558 of Mathematical Reviews Vol. , Issue 2002C [page]

2002 Mathematical Reviews  
(i-WPI; Worcester, MA); Szemerédi, Endre (1-RTG-MD; Piscataway, NJ) Proof of the Alon-Yuster conjecture. (English summary) Combinatorics (Prague, 1998). Discrete Math. 235 (2001), no. 1-3, 255-269.  ...  Summary: “In this paper we prove the following conjecture of N Alon and R. Yuster [J. Combin. Theory Ser. B 66 (1996), no. 2, 269-282; MR 97a:05162].  ... 

Page 3205 of Mathematical Reviews Vol. , Issue 2003e [page]

2003 Mathematical Reviews  
A common extension of the Erdés-Stone theorem and the Alon-Yuster theorem for unbounded graphs.  ...  A common extension of two fundamental results in extremal graph theory, namely the classical Erdds-Stone theorem and the more 05C Graph theory 2003e:05069 recent result of Alon-Yuster, is proved.  ... 

An average degree condition for independent transversals [article]

Stefan Glock, Benny Sudakov
2022 arXiv   pre-print
We can also use our new theorem to establish tight bounds for a more general version of the Erdős–Gyárfás–Łuczak problem and solve a conjecture of Yuster from 1997.  ...  This exploits a connection to the Turán numbers of complete bipartite graphs, which might be of independent interest.  ...  Whereas our main motivation came from the Erdős-Gyárfás-Luczak problem, they took the perspective of vertex colouring, and their proof is slightly different.  ... 
arXiv:2003.01683v2 fatcat:x2mfr7dtdfbh7m6e6lnbr4ouqa

Index to Volume 23

2002 European journal of combinatorics (Print)  
Theorem and the Alon-Yuster Theorem for Unbounded Graphs . . . . . 431 IVANOV, A.  ...  ., Proof of a Conjecture of Ehrenborg and Steingrímsson on Ex-ZHOU, S., Block Primitive 2-(v, k, 1) Designs Admitting a Ree Simple Group 1085 ZHOU, S., Constructing a Class of Symmetric Graphs . ., The  ... 
doi:10.1006/eujc.2002.0041 fatcat:77afpkkn35dbrnwpnndkb64tvi

Contents of Volume 23

2002 European journal of combinatorics (Print)  
T., The Terwilliger Algebra of the Hypercube . . . . . 399 ISHIGAMI, Y., A Common Extension of the Erdős-Stone Theorem and the Alon-Yuster Theorem for Unbounded Graphs . . . . . 431 MALNIČ  ...  L. and XIANG, Q., Proof of a Conjecture of De Caen and Van Dam . . . . . . . 201 MANTUROV, V.  ... 
doi:10.1006/eujc.2002.0040 fatcat:6zapgfe62naipplawwaptralda

A Common Extension of the Erdős–Stone Theorem and the Alon–Yuster Theorem for Unbounded Graphs

Yoshiyasu Ishigami
2002 European journal of combinatorics (Print)  
A Common Extension of the Erdős-Stone Theorem and the Alon-Yuster Theorem for Unbounded Graphs † YOSHIYASU ISHIGAMI The Erdős-Stone theorem (1946 , Bull. Am. Math.  ...  The important point is that our theorem enables us to deal with a larger graph H of order |H | → ∞ (as n → ∞), while |H | was fixed in the Alon-Yuster theorem (and in another common extension by Komlós  ...  On the other hand, the so-called Alon-Yuster theorem presents a sharp degree condition guaranteeing many disjoint copies of H as follows. THEOREM C ( [1] ).  ... 
doi:10.1006/eujc.2002.0575 fatcat:lra5cczx7jhmzhngm5xs6zdauy

A note on antimagic orientations of even regular graphs [article]

Donglei Yang
2019 arXiv   pre-print
Motivated by the conjecture of Hartsfield and Ringel on antimagic labelings of undirected graphs, Hefetz, Mütze, and Schwartz initiated the study of antimagic labelings of digraphs in 2010.  ...  Together with known results, our main result implies that the above-mentioned conjecture is true for all regular graphs.  ...  The author would like to thank Tong Li, Zi-Xia Song and Guanghui Wang for helpful discussion. This work was supported by the National Natural Science Foundation of China (11871311).  ... 
arXiv:1811.01904v2 fatcat:sgzsh3jjdzd7betkjc63puoomi

Subquadratic time approximation algorithms for the girth [chapter]

Liam Roditty, Virginia Vassilevska Williams
2012 Proceedings of the Twenty-Third Annual ACM-SIAM Symposium on Discrete Algorithms  
This result complements the work of Yuster and Zwick [SIAM J. Discrete Math'97] who showed how to compute g ′ in O(n 2 ) time. Multiplicative Approximations.  ...  g ′ is the length of the shortest even cycle in G.  ...  The color-coding portion of the algorithm can be efficiently derandomized, using the ideas of Alon, Yuster and Zwick [2] .  ... 
doi:10.1137/1.9781611973099.67 dblp:conf/soda/RodittyW12 fatcat:hwcabjmc6vde7nu3esual3qxmu

Rainbow connections of graphs -- A survey [article]

Xueliang Li, Yuefang Sun
2011 arXiv   pre-print
In this survey we attempt to bring together most of the results and papers that dealt with it.  ...  The concept of rainbow connection was introduced by Chartrand et al. in 2008. It is fairly interesting and recently quite a lot papers have been published about it.  ...  Algorithms and computational complexity At the end of [9] , Caro, Lev, Roditty, Tuza and Yuster gave two conjectures (see Conjecture 4.1 and Conjecture 4.2 in [9] ) on the complexity of determining the  ... 
arXiv:1101.5747v2 fatcat:zwwaavvzzjbwtjvn4wujnauibu

Large induced subgraphs with k vertices of almost maximum degree [article]

António Girão, Kamil Popielarz
2017 arXiv   pre-print
This solves a conjecture of Caro and Yuster up to the constant g_2(k).  ...  If G is a graph on n vertices with maximum degree Δ then it contains an induced subgraph H on at least n - g_1(k)√(Δ) vertices, such that H has k vertices of the same degree of order at least Δ(H)-g_2(  ...  In order to prove Theorem 1 we need the following theorem of Caro, Shapira and Yuster, appearing in [6] , whose proof is inspired by the one used by Alon and Berman in [1] . Theorem 5.  ... 
arXiv:1705.08998v1 fatcat:wqfbcevqjfexxodvaiv555xtfy

Page 4654 of Mathematical Reviews Vol. , Issue 2002G [page]

2002 Mathematical Reviews  
In this note, we give a much shorter proof of the result.  ...  The need for computing machinery in the proof is suggested by the decimal lengths of several weights: 12, 15, and 31 digits.  ... 

Counting graph orientations with no directed triangles [article]

Pedro Araújo, Fábio Botler, Guilherme Oliveira Mota
2020 arXiv   pre-print
Alon and Yuster proved that the number of orientations of any n-vertex graph in which every K_3 is transitively oriented is at most 2^ n^2/4 for n ≥ 10^4 and conjectured that the precise lower bound on  ...  We confirm their conjecture and, additionally, characterize the extremal families by showing that the balanced complete bipartite graph with n vertices is the only n-vertex graph for which there are exactly  ...  For related problems in the context of random graphs, the reader is referred to [1, 5] .  ... 
arXiv:2005.13091v1 fatcat:ljzwg2b27fb2hm25gzqpenk56m

Frugal, acyclic and star colourings of graphs

Ross J. Kang, Tobias Müller
2011 Discrete Applied Mathematics  
Given a graph G = (V , E), a vertex colouring of V is t-frugal if no colour appears more than t times in any neighbourhood and is acyclic if each of the bipartite graphs consisting of the edges between  ...  For graphs of bounded maximum degree, Hind et al. (1997) [14] studied proper t-frugal colourings and Yuster (1998) [22] studied acyclic proper 2-frugal colourings.  ...  We also appreciate the constructive comments of the anonymous referees. This work was initiated during a visit of RJK to EURANDOM. We gratefully acknowledge its support.  ... 
doi:10.1016/j.dam.2010.05.008 fatcat:hyftmnpxdvghdfvpesokxm356i
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