摘要
A k-colouring(not necessarily proper) of vertices of a graph is called acyclic, if for every pair of distinct colours i and j the subgraph induced by the edges whose endpoints have colours i and j is acyclic. We consider acyclic k-colourings such that each colour class induces a graph with a given(hereditary) property. In particular, we consider acyclic k-colourings in which each colour class induces a graph with maximum degree at most t, which are referred to as acyclic t-improper k-colourings. The acyclic t-improper chromatic number of a graph G is the smallest k for which there exists an acyclic t-improper k-colouring of G. We focus on acyclic colourings of graphs with maximum degree 4. We prove that 3 is an upper bound for the acyclic 3-improper chromatic number of this class of graphs. We also provide a non-trivial family of graphs with maximum degree4 whose acyclic 3-improper chromatic number is at most 2, namely, the graphs with maximum average degree at most 3. Finally, we prove that any graph G with Δ(G) 4 can be acyclically coloured with 4 colours in such a way that each colour class induces an acyclic graph with maximum degree at most 3.
A k-colouring (not necessarily proper) of vertices of a graph is called acyclic, if for every pair of distinct colours i and j the subgraph induced by the edges whose endpoints have colours i and j is acyclic. We consider acyclie k-eolourings such that each colour class induces a graph with a given (hereditary) property. In particular, we consider aeyclic k-eolourings in which each colour class induces a graph with maximum degree at most t, which are referred to as acyclic t-improper k-colourings. The acyelic t-improper chromatic number of a graph G is the smallest k for which there exists an acyclic t-improper k-colouring of G. We focus on acyclic colourings of graphs with maximum degree 4. We prove that 3 is an upper bound for the acyclic 3-improper chromatic number of this class of graphs. We also provide a non-trivial family of graphs with maximum degree 4 whose acyclic 3-improper chromatic number is at most 2, namely, the graphs with maximum average degree at most 3. Finally, we prove that any graph G with A(G) ≤ 4 can be acyelically coloured with 4 colours in such a way that each colour class induces an acyclic graph with maximum degree at most 3.
基金
supported by the Minister of Science and Higher Education of Poland(Grant No.JP2010009070)
关键词
非周期性
最大度
着色
图表
特性曲线
无环
颜色
诱导
acyclic colouring, acyclic improper colouring, bounded degree graph, hereditary property