Homogeneous sets, clique-separators, critical graphs, and optimal χ\chi-binding functions

Given a set $\mathcal{H}$ of graphs, let $f_\mathcal{H}^\star\colon \mathbb{N}_{>0}\to \mathbb{N}_{>0}$ be the optimal $\chi$-binding function of the class of $\mathcal{H}$-free graphs, that is, $$f_\mathcal{H}^\star(\omega)=\max\{\chi(G): G\text{ is } \mathcal{H}\text{-free, } \omega(G)=\omega\}.$$ In this paper, we combine the two decomposition methods by homogeneous sets and clique-separators in order to determine optimal $\chi$-binding functions for subclasses of $P_5$-free graphs and of $(C_5,C_7,\ldots)$-free graphs. In particular, we prove the following for each $\omega\geq 1$: (i) $\ f_{\{P_5,banner\}}^\star(\omega)=f_{3K_1}^\star(\omega)\in \Theta(\omega^2/\log(\omega)),$ (ii) $\ f_{\{P_5,co-banner\}}^\star(\omega)=f^\star_{\{2K_2\}}(\omega)\in\mathcal{O}(\omega^2),$(iii)$\ f_{\{C_5,C_7,\ldots,banner\}}^\star(\omega)=f^\star_{\{C_5,3K_1\}}(\omega)\notin \mathcal{O}(\omega),$and (iv)$\ f_{\{P_5,C_4\}}^\star(\omega)=\lceil(5\omega-1)/4\rceil.$We also characterise, for each of our considered graph classes, all graphs$G$with$\chi(G)>\chi(G-u)$for each$u\in V(G)$. From these structural results, we can prove Reed's conjecture -- relating chromatic number, clique number, and maximum degree of a graph -- for$(P_5,banner)$-free graphs

The Price of Connectivity for Vertex Cover

The vertex cover number of a graph is the minimum number of vertices that are needed to cover all edges. When those vertices are further required to induce a connected subgraph, the corresponding number is called the connected vertex cover number, and is always greater or equal to the vertex cover number. Connected vertex covers are found in many applications, and the relationship between those two graph invariants is therefore a natural question to investigate. For that purpose, we introduce the {\em Price of Connectivity}, defined as the ratio between the two vertex cover numbers. We prove that the price of connectivity is at most 2 for arbitrary graphs. We further consider graph classes in which the price of connectivity of every induced subgraph is bounded by some real number $t$. We obtain forbidden induced subgraph characterizations for every real value $t \leq 3/2$. We also investigate critical graphs for this property, namely, graphs whose price of connectivity is strictly greater than that of any proper induced subgraph. Those are the only graphs that can appear in a forbidden subgraph characterization for the hereditary property of having a price of connectivity at most $t$. In particular, we completely characterize the critical graphs that are also chordal. Finally, we also consider the question of computing the price of connectivity of a given graph. Unsurprisingly, the decision version of this question is NP-hard. In fact, we show that it is even complete for the class $\Theta_2^P = P^{NP[\log]}$, the class of decision problems that can be solved in polynomial time, provided we can make $O(\log n)$ queries to an NP-oracle. This paves the way for a thorough investigation of the complexity of problems involving ratios of graph invariants.Comment: 19 pages, 8 figure

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

Small clique number graphs with three trivial critical ideals

The critical ideals of a graph are the determinantal ideals of the generalized Laplacian matrix associated to a graph. In this article we provide a set of minimal forbidden graphs for the set of graphs with at most three trivial critical ideals. Then we use these forbidden graphs to characterize the graphs with at most three trivial critical ideals and clique number equal to 2 and 3.Comment: 33 pages, 3 figure