TY - JOUR

T1 - Rainbow trees in uniformly edge-colored graphs

AU - Aigner-Horev, Elad

AU - Hefetz, Dan

AU - Lahiri, Abhiruk

N1 - Publisher Copyright:
© 2022 Wiley Periodicals LLC.

PY - 2022/6/17

Y1 - 2022/6/17

N2 - We obtain sufficient conditions for the emergence of spanning and almost-spanning bounded-degree rainbow trees in various host graphs, having their edges colored independently and uniformly at random, using a predetermined palette. Our first result asserts that a uniform coloring of (Formula presented.), using a palette of size (Formula presented.), a.a.s. admits a rainbow copy of any given bounded-degree tree on at most (Formula presented.) vertices, where (Formula presented.) is arbitrarily small yet fixed. This serves as a rainbow variant of a classical result by Alon et al. pertaining to the embedding of bounded-degree almost-spanning prescribed trees in (Formula presented.), where (Formula presented.) is independent of (Formula presented.). Given an (Formula presented.) -vertex graph (Formula presented.) with minimum degree at least (Formula presented.), where (Formula presented.) is fixed, we use our aforementioned result in order to prove that a uniform coloring of the randomly perturbed graph (Formula presented.), using (Formula presented.) colors, where (Formula presented.) is arbitrarily small yet fixed, a.a.s. admits a rainbow copy of any given bounded-degree spanning tree. This can be viewed as a rainbow variant of a result by Krivelevich et al. who proved that (Formula presented.), where (Formula presented.) is independent of (Formula presented.), a.a.s. admits a copy of any given bounded-degree spanning tree. Finally, and with (Formula presented.) as above, we prove that a uniform coloring of (Formula presented.) using (Formula presented.) colors a.a.s. admits a rainbow spanning tree. Put another way, the trivial lower bound on the size of the palette required for supporting a rainbow spanning tree is also sufficient, essentially as soon as the random perturbation a.a.s. has edges.

AB - We obtain sufficient conditions for the emergence of spanning and almost-spanning bounded-degree rainbow trees in various host graphs, having their edges colored independently and uniformly at random, using a predetermined palette. Our first result asserts that a uniform coloring of (Formula presented.), using a palette of size (Formula presented.), a.a.s. admits a rainbow copy of any given bounded-degree tree on at most (Formula presented.) vertices, where (Formula presented.) is arbitrarily small yet fixed. This serves as a rainbow variant of a classical result by Alon et al. pertaining to the embedding of bounded-degree almost-spanning prescribed trees in (Formula presented.), where (Formula presented.) is independent of (Formula presented.). Given an (Formula presented.) -vertex graph (Formula presented.) with minimum degree at least (Formula presented.), where (Formula presented.) is fixed, we use our aforementioned result in order to prove that a uniform coloring of the randomly perturbed graph (Formula presented.), using (Formula presented.) colors, where (Formula presented.) is arbitrarily small yet fixed, a.a.s. admits a rainbow copy of any given bounded-degree spanning tree. This can be viewed as a rainbow variant of a result by Krivelevich et al. who proved that (Formula presented.), where (Formula presented.) is independent of (Formula presented.), a.a.s. admits a copy of any given bounded-degree spanning tree. Finally, and with (Formula presented.) as above, we prove that a uniform coloring of (Formula presented.) using (Formula presented.) colors a.a.s. admits a rainbow spanning tree. Put another way, the trivial lower bound on the size of the palette required for supporting a rainbow spanning tree is also sufficient, essentially as soon as the random perturbation a.a.s. has edges.

KW - edge-coloring

KW - rainbow trees

KW - random graphs

KW - random perturbation

UR - http://www.scopus.com/inward/record.url?scp=85131942808&partnerID=8YFLogxK

U2 - 10.1002/rsa.21103

DO - 10.1002/rsa.21103

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AN - SCOPUS:85131942808

SN - 1042-9832

VL - 62

SP - 287

EP - 303

JO - Random Structures and Algorithms

JF - Random Structures and Algorithms

IS - 2

ER -