TY - JOUR
T1 - Coloring Graphs From Random Lists
AU - Hefetz, Dan
AU - Krivelevich, Michael
N1 - Publisher Copyright:
© 2025 Wiley Periodicals LLC.
PY - 2025/1
Y1 - 2025/1
N2 - Given positive integers (Formula presented.) and a graph (Formula presented.), a family of lists (Formula presented.) is said to be a random (Formula presented.) -list-assignment if for every (Formula presented.) the list (Formula presented.) is a subset of (Formula presented.) of size (Formula presented.), chosen uniformly at random and independently of the choices of all other vertices. An (Formula presented.) -vertex graph (Formula presented.) is said to be a.a.s. (Formula presented.) -colorable if (Formula presented.), where (Formula presented.) is a random (Formula presented.) -list-assignment. We prove that if (Formula presented.) and (Formula presented.), where (Formula presented.) is the maximum degree of (Formula presented.) and (Formula presented.) is an integer, then (Formula presented.) is a.a.s. (Formula presented.) -colorable. This is not far from being the best possible, forms a continuation of the so-called palette sparsification results, and proves in a strong sense a conjecture of Casselgren. Furthermore, we consider this problem under the additional assumption that (Formula presented.) is (Formula presented.) -free for some graph (Formula presented.). For various graphs (Formula presented.), we estimate the smallest (Formula presented.) for which any (Formula presented.) -free (Formula presented.) -vertex graph (Formula presented.) is a.a.s. (Formula presented.) -colorable for every (Formula presented.). This extends and improves several results of Casselgren.
AB - Given positive integers (Formula presented.) and a graph (Formula presented.), a family of lists (Formula presented.) is said to be a random (Formula presented.) -list-assignment if for every (Formula presented.) the list (Formula presented.) is a subset of (Formula presented.) of size (Formula presented.), chosen uniformly at random and independently of the choices of all other vertices. An (Formula presented.) -vertex graph (Formula presented.) is said to be a.a.s. (Formula presented.) -colorable if (Formula presented.), where (Formula presented.) is a random (Formula presented.) -list-assignment. We prove that if (Formula presented.) and (Formula presented.), where (Formula presented.) is the maximum degree of (Formula presented.) and (Formula presented.) is an integer, then (Formula presented.) is a.a.s. (Formula presented.) -colorable. This is not far from being the best possible, forms a continuation of the so-called palette sparsification results, and proves in a strong sense a conjecture of Casselgren. Furthermore, we consider this problem under the additional assumption that (Formula presented.) is (Formula presented.) -free for some graph (Formula presented.). For various graphs (Formula presented.), we estimate the smallest (Formula presented.) for which any (Formula presented.) -free (Formula presented.) -vertex graph (Formula presented.) is a.a.s. (Formula presented.) -colorable for every (Formula presented.). This extends and improves several results of Casselgren.
KW - graph colouring
KW - list colouring
KW - palette sparsification
KW - random lists
UR - http://www.scopus.com/inward/record.url?scp=85215546345&partnerID=8YFLogxK
U2 - 10.1002/rsa.21278
DO - 10.1002/rsa.21278
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AN - SCOPUS:85215546345
SN - 1042-9832
VL - 66
JO - Random Structures and Algorithms
JF - Random Structures and Algorithms
IS - 1
M1 - e21278
ER -