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تسجيل مستخدم جديدChoosability with union separation
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List coloring generalizes graph coloring by requiring the color of a vertex to be selected from a list of colors specific to that vertex. One refinement of list coloring, called choosability with separation, requires that the intersection of adjacent lists is sufficiently small. We introduce a new refinement, called choosability with union separation, where we require that the union of adjacent lists is sufficiently large. For $t geq k$, a $(k,t)$-list assignment is a list assignment $L$ where $|L(v)| geq k$ for all vertices $v$ and $|L(u)cup L(v)| geq t$ for all edges $uv$. A graph is $(k,t)$-choosable if there is a proper coloring for every $(k,t)$-list assignment. We explore this concept through examples of graphs that are not $(k,t)$-choosable, demonstrating sparsity conditions that imply a graph is $(k,t)$-choosable, and proving that all planar graphs are $(3,11)$-choosable and $(4,9)$-choosable.
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All planar graphs are 4-colorable and 5-choosable, while some planar graphs are not 4-choosable. Determining which properties guarantee that a planar graph can be colored using lists of size four has received significant attention. In terms of constr
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Assume $ k $ is a positive integer, $ lambda={k_1,k_2,...,k_q} $ is a partition of $ k $ and $ G $ is a graph. A $lambda$-assignment of $ G $ is a $ k $-assignment $ L $ of $ G $ such that the colour set $ bigcup_{vin V(G)} L(v) $ can be partitioned
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