Equitable defective coloring of sparse planar graphs
نویسندگان
چکیده
A graph has an equitable, defective k-coloring (an ED-k-coloring) if there is a kcoloring of V (G) that is defective (every vertex shares the same color with at most one neighbor) and equitable (the sizes of all color classes differ by at most one). A graph may have an ED-kcoloring, but no ED-(k + 1)-coloring. In this paper, we prove that planar graphs with minimum degree at least 2 and girth at least 10 are ED-k-colorable for any integer k ≥ 3. The proof uses the method of discharging. We are able to simplify the normally lengthy task of enumerating forbidden substructures by using Hall’s Theorem, an unusual approach.
منابع مشابه
Equitable Coloring of Sparse Planar Graphs
A proper vertex coloring of a graph G is equitable if the sizes of color classes differ by at most one. The equitable chromatic threshold χeq(G) of G is the smallest integer m such that G is equitably n-colorable for all n ≥ m. We show that for planar graphs G with minimum degree at least two, χeq(G) ≤ 4 if the girth of G is at least 10, and χeq(G) ≤ 3 if the girth of G is at least 14.
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عنوان ژورنال:
- Discrete Mathematics
دوره 312 شماره
صفحات -
تاریخ انتشار 2012