A clique of a graph G is a set of pairwise adjacent vertices of G. A clique-coloring of G is an assignment of colors to the vertices of G in such a way that no inclusion-wise maximal clique of size at least two of G is monochromatic. An equitable clique-coloring of G is a clique-coloring such that any two color classes differ in size by at most one. Bacsó and Tuza proved that connected claw-free graphs with maximum degree at most four, other than chordless odd cycles of order greater than three, are 2-clique-colorable and a 2-clique-coloring can be found in \(O(n^{2})\) Bacsó and Tuza (Discrete Math Theor Comput Sci 11(2):15–24, 2009). In this paper we prove that every connected claw-free graph with maximum degree at most four, not a chordless odd cycle of order greater than three, has an equitable 2-clique-coloring. In addition we improve the algorithm described in the paper mentioned by giving an equitable 2-clique-coloring in linear time for this class of graphs.

甲集团的曲线图的ģ是一组两两相邻顶点的ģ。甲集团着色的ģ是颜色的顶点的分配ģ在这样一种方式，尺寸的夹杂物没有逐极大团中的至少两个ģ是单色的。一个公平集团着色的ģ是一个集团着色，使得任何两个颜色的类尺寸至多一个相差。Bacsó和Tuza证明，最大程度为4的连通无爪图，除了大于3的无弦奇数环之外，都是2色可着色的，并且在\（O（n ^ {2}）\）Bacsó和Tuza（离散数学理论计算科学11（2）：15-24，2009年）。在本文中，我们证明每一个最大度为4的连通无爪图，不是大于3的无弦奇数环，都具有相等的2色着色。另外，我们通过为此类图在线性时间内给出相等的2-clicli-coloring来改进本文所述的算法。