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arXiv: 2110.05230
AbstractList coloring is an influential and classic topic in graph theory. We initiate the study of a natural strengthening of this problem, where instead of one list‐coloring, we seek many in parallel. Our explorations have uncovered a potentially rich seam of interesting problems spanning chromatic graph theory. Given a ‐list‐assignment of a graph , which is the assignment of a list of colors to each vertex , we study the existence of pairwise‐disjoint proper colorings of using colors from these lists. We may refer to this as a list‐packing. Using a mix of combinatorial and probabilistic methods, we set out some basic upper bounds on the smallest for which such a list‐packing is always guaranteed, in terms of the number of vertices, the degeneracy, the maximum degree, or the (list) chromatic number of . (The reader might already find it interesting that such a minimal is well defined.) We also pursue a more focused study of the case when is a bipartite graph. Our results do not yet rule out the tantalising prospect that the minimal above is not too much larger than the list chromatic number. Our study has taken inspiration from study of the strong chromatic number, and we also explore generalizations of the problem above in the same spirit.
FOS: Computer and information sciences, Technology, Discrete Mathematics (cs.DM), TRANSVERSALS, Mathematics, Applied, 511, GRAPHS, 0101 Pure Mathematics, Computation Theory & Mathematics, NUMBER, Coloring of graphs and hypergraphs, Edge subsets with special properties (factorization, matching, partitioning, covering and packing, etc.), Transversal (matching) theory, FOS: Mathematics, Mathematics - Combinatorics, 4901 Applied mathematics, list colouring, 05C15, 05C35, 05C69, 05C70, graph packing, 0802 Computation Theory and Mathematics, 4613 Theory of computation, graph colouring, Science & Technology, 0104 Statistics, Computer Science, Software Engineering, 004, Computer Science, Physical Sciences, 4904 Pure mathematics, Combinatorics (math.CO), strong chromatic number, Mathematics, independent transversals, Computer Science - Discrete Mathematics
FOS: Computer and information sciences, Technology, Discrete Mathematics (cs.DM), TRANSVERSALS, Mathematics, Applied, 511, GRAPHS, 0101 Pure Mathematics, Computation Theory & Mathematics, NUMBER, Coloring of graphs and hypergraphs, Edge subsets with special properties (factorization, matching, partitioning, covering and packing, etc.), Transversal (matching) theory, FOS: Mathematics, Mathematics - Combinatorics, 4901 Applied mathematics, list colouring, 05C15, 05C35, 05C69, 05C70, graph packing, 0802 Computation Theory and Mathematics, 4613 Theory of computation, graph colouring, Science & Technology, 0104 Statistics, Computer Science, Software Engineering, 004, Computer Science, Physical Sciences, 4904 Pure mathematics, Combinatorics (math.CO), strong chromatic number, Mathematics, independent transversals, Computer Science - Discrete Mathematics
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