Infinite families of kk-vertex-critical (P5P_5, C5C_5)-free graphs

A graph is $k$-vertex-critical if $\chi(G)=k$ but $\chi(G-v)<k$ for all $v\in V(G)$. We construct a new infinite families of $k$-vertex-critical $(P_5,C_5)$-free graphs for all $k\ge 6$. Our construction generalizes known constructions for $4$-vertex-critical $P_7$-free graphs and $5$-vertex-critical $P_5$-free graphs and is in contrast to the fact that there are only finitely many $5$-vertex-critical $(P_5,C_5)$-free graphs. In fact, our construction is actually even more well-structured, being $(2P_2,K_3+P_1,C_5)$-free

Exhaustive generation of kk-critical H\mathcal H-free graphs

We describe an algorithm for generating all $k$-critical $\mathcal H$-free graphs, based on a method of Ho\`{a}ng et al. Using this algorithm, we prove that there are only finitely many $4$-critical $(P_7,C_k)$-free graphs, for both $k=4$ and $k=5$. We also show that there are only finitely many $4$-critical graphs $(P_8,C_4)$-free graphs. For each case of these cases we also give the complete lists of critical graphs and vertex-critical graphs. These results generalize previous work by Hell and Huang, and yield certifying algorithms for the $3$-colorability problem in the respective classes. Moreover, we prove that for every $t$, the class of 4-critical planar $P_t$-free graphs is finite. We also determine all 27 4-critical planar $(P_7,C_6)$-free graphs. We also prove that every $P_{10}$-free graph of girth at least five is 3-colorable, and determine the smallest 4-chromatic $P_{12}$-free graph of girth five. Moreover, we show that every $P_{13}$-free graph of girth at least six and every $P_{16}$-free graph of girth at least seven is 3-colorable. This strengthens results of Golovach et al.Comment: 17 pages, improved girth results. arXiv admin note: text overlap with arXiv:1504.0697

A refinement on the structure of vertex-critical (P5P_5, gem)-free graphs

We give a new, stronger proof that there are only finitely many $k$-vertex-critical ($P_5$,~gem)-free graphs for all $k$. Our proof further refines the structure of these graphs and allows for the implementation of a simple exhaustive computer search to completely list all $6$- and $7$-vertex-critical $(P_5$, gem)-free graphs. Our results imply the existence of polynomial-time certifying algorithms to decide the $k$-colourability of $(P_5$, gem)-free graphs for all $k$ where the certificate is either a $k$-colouring or a $(k+1)$-vertex-critical induced subgraph. Our complete lists for $k\le 7$ allow for the implementation of these algorithms for all $k\le 6$

Critical (P5,dart)(P_5,dart)-Free Graphs

Given two graphs $H_1$ and $H_2$, a graph is $(H_1,H_2)$-free if it contains no induced subgraph isomorphic to $H_1$ nor $H_2$. Let $P_t$ be the path on $t$ vertices. A dart is the graph obtained from a diamond by adding a new vertex and making it adjacent to exactly one vertex with degree 3 in the diamond. In this paper, we show that there are finitely many $k$-vertex-critical $(P_5,dart)$-free graphs for $k \ge 1$ To prove these results, we use induction on $k$ and perform a careful structural analysis via Strong Perfect Graph Theorem combined with the pigeonhole principle based on the properties of vertex-critical graphs. Moreover, for $k \in \{5, 6, 7\}$ we characterize all $k$-vertex-critical $(P_5,dart)$-free graphs using a computer generation algorithm. Our results imply the existence of a polynomial-time certifying algorithm to decide the $k$-colorability of $(P_5,dart)$-free graphs for $k \ge 1$ where the certificate is either a $k$-coloring or a $(k+1)$-vertex-critical induced subgraph.Comment: arXiv admin note: text overlap with arXiv:2211.0417

Critical (P5P_5,bull)-free graphs

Given two graphs $H_1$ and $H_2$, a graph is $(H_1,H_2)$-free if it contains no induced subgraph isomorphic to $H_1$ or $H_2$. Let $P_t$ and $C_t$ be the path and the cycle on $t$ vertices, respectively. A bull is the graph obtained from a triangle with two disjoint pendant edges. In this paper, we show that there are finitely many 5-vertex-critical ($P_5$,bull)-free graphs.Comment: 21 page