On r-hued coloring of planar graphs with girth at least 6
نویسندگان
چکیده
For integers k, r > 0, a (k, r)-coloring of a graph G is a proper k-coloring c such that for any vertex v with degree d(v), v is adjacent to at least min{d(v), r} different colors. Such coloring is also called as an r-hued coloring. The r-hued chromatic number of G, χr (G), is the least integer k such that a (k, r)-coloring of G exists. In this paper, we proved that if G is a planar graph with girth at least 6, then χr (G) ≤ r + 5. This extends a former result in Bu and Zhu (2012). It also implies that a conjecture on r-hued coloring of planar graphs is true for planar graphs with girth at least 6. © 2015 Elsevier B.V. All rights reserved.
منابع مشابه
List coloring the square of sparse graphs with large degree
We consider the problem of coloring the squares of graphs of bounded maximum average degree, that is, the problem of coloring the vertices while ensuring that two vertices that are adjacent or have a common neighbour receive different colors. Borodin et al. proved in 2004 and 2008 that the squares of planar graphs of girth at least seven and sufficiently large maximum degree ∆ are list (∆ + 1)-...
متن کاملSplitting Planar Graphs of Girth 6 into Two Linear Forests with Short Paths
Recently, Borodin, Kostochka, and Yancey (On 1-improper 2-coloring of sparse graphs. Discrete Mathematics, 313(22), 2013) showed that the vertices of each planar graph of girth at least 7 can be 2-colored so that each color class induces a subgraph of a matching. We prove that any planar graph of girth at least 6 admits a vertex coloring in 2 colors such that each monochromatic component is a p...
متن کاملAdjacent vertex distinguishing edge-colorings of planar graphs with girth at least six
An adjacent vertex distinguishing edge-coloring of a graph G is a proper edge-coloring of G such that any pair of adjacent vertices are incident to distinct sets of colors. The minimum number of colors required for an adjacent vertex distinguishing edge-coloring of G is denoted by χ a (G). We prove that χ a (G) is at most the maximum degree plus 2 if G is a planar graph without isolated edges w...
متن کامل2 - distance ( ∆ + 2 ) - coloring of planar graphs with girth six and ∆ ≥ 18
It was proved in [Z. Dvořàk, D. Kràl, P. Nejedlỳ, R. Škrekovski, Coloring squares of planar graphs with girth six, European J. Combin. 29 (4) (2008) 838–849] that every planar graph with girth g ≥ 6 and maximum degree ∆ ≥ 8821 is 2-distance (∆ + 2)-colorable. We prove that every planar graph with g ≥ 6 and∆ ≥ 18 is 2-distance (∆+ 2)-colorable. © 2009 Elsevier B.V. All rights reserved.
متن کاملInjective choosability of subcubic planar graphs with girth 6
An injective coloring of a graph G is an assignment of colors to the vertices of G so that any two vertices with a common neighbor have distinct colors. A graph G is injectively k-choosable if for any list assignment L, where |L(v)| ≥ k for all v ∈ V (G), G has an injective L-coloring. Injective colorings have applications in the theory of error-correcting codes and are closely related to other...
متن کاملذخیره در منابع من
با ذخیره ی این منبع در منابع من، دسترسی به آن را برای استفاده های بعدی آسان تر کنید
عنوان ژورنال:
- Discrete Applied Mathematics
دوره 198 شماره
صفحات -
تاریخ انتشار 2016