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Proof of the Alon–Yuster conjecture

2001
*
Discrete Mathematics
*

In this paper we prove

doi:10.1016/s0012-365x(00)00279-x
fatcat:omiqo3umubbajdkuee33r74py4
*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'). ...##
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$K_{\ell}^{-}$-factors in graphs

2005
*
Discrete Mathematics & Theoretical Computer Science
*

This is best possible up to

doi:10.46298/dmtcs.3403
fatcat:7hfdtpeitrgl5o2mptkxrklqgm
*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. ...##
###
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. ...##
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An average degree condition for independent transversals
[article]

2022
*
arXiv
*
pre-print

We can also use our new theorem to establish tight bounds for a more general version

arXiv:2003.01683v2
fatcat:x2mfr7dtdfbh7m6e6lnbr4ouqa
*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. ...##
###
Index to Volume 23

2002
*
European journal of combinatorics (Print)
*

Theorem and

doi:10.1006/eujc.2002.0041
fatcat:77afpkkn35dbrnwpnndkb64tvi
*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*...##
###
Contents of Volume 23

2002
*
European journal of combinatorics (Print)
*

T.,

doi:10.1006/eujc.2002.0040
fatcat:6zapgfe62naipplawwaptralda
*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. ...##
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A Common Extension of the Erdős–Stone Theorem and the Alon–Yuster Theorem for Unbounded Graphs

2002
*
European journal of combinatorics (Print)
*

A Common Extension

doi:10.1006/eujc.2002.0575
fatcat:lra5cczx7jhmzhngm5xs6zdauy
*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] ). ...##
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A note on antimagic orientations of even regular graphs
[article]

2019
*
arXiv
*
pre-print

Motivated by

arXiv:1811.01904v2
fatcat:sgzsh3jjdzd7betkjc63puoomi
*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). ...##
###
Subquadratic time approximation algorithms for the girth
[chapter]

2012
*
Proceedings of the Twenty-Third Annual ACM-SIAM Symposium on Discrete Algorithms
*

This result complements

doi:10.1137/1.9781611973099.67
dblp:conf/soda/RodittyW12
fatcat:hwcabjmc6vde7nu3esual3qxmu
*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] . ...##
###
Rainbow connections of graphs -- A survey
[article]

2011
*
arXiv
*
pre-print

In this survey we attempt to bring together most

arXiv:1101.5747v2
fatcat:zwwaavvzzjbwtjvn4wujnauibu
*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*...##
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Large induced subgraphs with k vertices of almost maximum degree
[article]

2017
*
arXiv
*
pre-print

This solves a

arXiv:1705.08998v1
fatcat:wqfbcevqjfexxodvaiv555xtfy
*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. ...##
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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. ...##
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Counting graph orientations with no directed triangles
[article]

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] . ...

##
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Frugal, acyclic and star colourings of graphs

2011
*
Discrete Applied Mathematics
*

Given a graph G = (V , E), a vertex colouring

doi:10.1016/j.dam.2010.05.008
fatcat:hyftmnpxdvghdfvpesokxm356i
*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. ...
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