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Upper Bounds of Dynamic Chromatic Number.
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The best possible upper bounds as an analogue to the Brook’s Theorem are proved, together with the determination of chromatic numbers for complete k-partite graphs.Abstract:
A proper vertex k-coloring of a graph G is dynamic if for every vertex v with degree at least 2, the neighbors of v receive at least two different colors. The smallest integer k such that G has a dynamic k-coloring is the dynamic chromatic number χd(G). We prove in this paper the following best possible upper bounds as an analogue to the Brook’s Theorem, together with the determination of chromatic numbers for complete k-partite graphs. (1) If ∆ ≤ 3, then χd(G) ≤ 4, with the only exception that G = C5, in which case χd(C5) = 5. (2) If ∆ ≥ 4, then χd(G) ≤ ∆+ 1. (3) χd(K1,1) = 2, χd(K1,m) = 3 and χd(Km,n) = 4 for m,n ≥ 2; χd(Kn1,n2,···,nk) = k for k ≥ 3.read more
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On the list dynamic coloring of graphs
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Dynamic coloring and list dynamic coloring of planar graphs
TL;DR: It is shown that ch"d(G)@?5 for every planar graph, which is the least number k such that for any assignment of k-element lists to the vertices of G, there is a dynamic coloring of G where the color on each vertex is chosen from its list.
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Note: Conditional colorings of graphs
TL;DR: The behavior and bounds of conditional chromatic number of a graph G are investigated and it is found that every vertex of degree at least r in G will be adjacent to vertices with at leastR different colors.
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On r -dynamic coloring of grids
TL;DR: It is shown that the m -by- n grid has no 3-dynamic 4-coloring when m n ? 2 mod 4 (for m , n ? 3 ), which completes the determination of the r -dynamic chromatic number of the m-by-N grid for all r, m, n.
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On dynamic coloring for planar graphs and graphs of higher genus
TL;DR: The dynamic chromatic number, denoted by @g"2(G), is the smallest integer k for which a graph G has a (k,2)-coloring, which is the least integer k such that every list assignment L with |L(v)|=k, @[email protected]?V(G], permits an (L,2-coloring.
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Book
Graph theory with applications
TL;DR: In this paper, the authors present Graph Theory with Applications: Graph theory with applications, a collection of applications of graph theory in the field of Operational Research and Management. Journal of the Operational research Society: Vol. 28, Volume 28, issue 1, pp. 237-238.