Three-coloring triangle-free graphs without long forbidden paths
Yidong Zhou, Jorik Jooken, Baoyuan Shan, Jan Goedgebeur, Shenwei Huang

TL;DR
This paper characterizes certain critical graphs and coloring properties in triangle-free graphs with forbidden paths, settling conjectures and constructing infinite families of critical graphs.
Contribution
It proves the uniqueness of three 4-vertex-critical graphs with specific forbidden subgraphs and characterizes the chromatic number for classes of graphs with forbidden induced subgraphs.
Findings
Exactly three 4-vertex-critical {P_7,C_3}- free graphs with an induced {C_7}.
All {P_5+P_1,C_3}- free graphs are 3-colorable.
Constructed an infinite family of 4-vertex-critical {4K_2,C_3}- free graphs.
Abstract
A graph is -vertex-critical if , but for every proper induced subgraph of . For a family of graphs , is -free if no graph is an induced subgraph of . We show that there are exactly three 4-vertex-critical -free graphs containing an induced , thereby settling the first of the two cases of a conjecture by Goedgebeur and Schaudt [J.~Graph Theory, 87:188--207, 2018]. Moreover, we show that all -free graphs are -colorable and by combining our result with known results from the literature, we completely characterize the maximum chromatic number of -free graphs if is a six-vertex induced subgraph of . Finally, we construct an infinite family of -vertex-critical -free graphs. These graphs are also -free and this is…
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Taxonomy
TopicsAdvanced Graph Theory Research · Limits and Structures in Graph Theory · Computational Geometry and Mesh Generation
