Problem
Research questions and friction points this paper is trying to address.
b-Chromatic Number
Tight b-Chromatic Number
Fall Chromatic Number
Fall Achromatic Number
H-free graphs
Optimal b-Colourings and Fall Colourings in $H$-Free Graphs
🤖 AI Summary
This study investigates the computational complexity of four graph coloring problems—b-chromatic number, tight b-chromatic number, fall chromatic number, and fall achromatic number—restricted to $H$-free graphs, i.e., graphs excluding a fixed graph $H$ as an induced subgraph. Through structural graph analysis, complexity reductions, and combinatorial constructions, the work systematically delineates the complexity landscape of these parameters across all $H$-free graph classes. A key contribution is the first identification of a graph $H$ for which computing the b-chromatic number is NP-hard while the tight b-chromatic number is solvable in polynomial time. Furthermore, the paper introduces a general technique for determining the complexity of the tight b-chromatic number, substantially expanding the known classes of graphs where this problem is either tractable or intractable.
📝 Abstract
In a colouring of a graph, a vertex is b-chromatic if it is adjacent to a vertex of every other colour. We consider four well-studied colouring problems: b-Chromatic Number, Tight b-Chromatic Number, Fall Chromatic Number and Fall Achromatic Number, which fit into a framework based on whether every colour class has (i) at least one b-chromatic vertex, (ii) exactly one b-chromatic vertex, or (iii) all of its vertices being b-chromatic. By combining known and new results, we fully classify the computational complexity of b-Chromatic Number, Fall Chromatic Number and Fall Achromatic Number in $H$-free graphs. For Tight b-Chromatic Number in $H$-free graphs, we develop a general technique to determine new graphs $H$, for which the problem is polynomial-time solvable, and we also determine new graphs $H$, for which the problem is still NP-complete. We show, for the first time, the existence of a graph $H$ such that in $H$-free graphs, b-Chromatic Number is NP-hard, while Tight b-Chromatic Number is polynomial-time solvable.
Innovation
Methods, ideas, or system contributions that make the work stand out.
b-chromatic number
fall colouring
H-free graphs
computational complexity
tight b-chromatic number
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