On graphs with no induced subdivision of K4K_4

We prove a decomposition theorem for graphs that do not contain a subdivision of $K_4$ as an induced subgraph where $K_4$ is the complete graph on four vertices. We obtain also a structure theorem for the class $\cal C$ of graphs that contain neither a subdivision of $K_4$ nor a wheel as an induced subgraph, where a wheel is a cycle on at least four vertices together with a vertex that has at least three neighbors on the cycle. Our structure theorem is used to prove that every graph in $\cal C$ is 3-colorable and entails a polynomial-time recognition algorithm for membership in $\cal C$. As an intermediate result, we prove a structure theorem for the graphs whose cycles are all chordless

Triangle-free graphs that do not contain an induced subdivision of K₄ are 3-colorable

We show that triangle-free graphs that do not contain an induced subgraph isomorphic to a subdivision of K4 are 3-colorable. This proves a conjecture of Trotignon and Vušković [J. Graph Theory. 84 (2017), no. 3, pp. 233–248]

A tight linear chromatic bound for (P3∪P2,W4P_3\cup P_2, W_4)-free graphs

For two vertex disjoint graphs $H$ and $F$, we use $H\cup F$ to denote the graph with vertex set $V(H)\cup V(F)$ and edge set $E(H)\cup E(F)$, and use $H+F$ to denote the graph with vertex set $V(H)\cup V(F)$ and edge set $E(H)\cup E(F)\cup\{xy\;|\; x\in V(H), y\in V(F)$$\}$. A $W_4$ is the graph $K_1+C_4$. In this paper, we prove that $\chi(G)\le 2\omega(G)$ if $G$ is a ($P_3\cup P_2, W_4$)-free graph. This bound is tight when $\omega =2$ and $3$, and improves the main result of Wang and Zhang. Also, this bound partially generalizes some results of Prashant {\em et al.}.Comment: arXiv admin note: text overlap with arXiv:2308.05442, arXiv:2307.1194

On triangle-free graphs that do not contain a subdivision of the complete graph on four vertices as an induced subgraph

We prove a decomposition theorem for the class of triangle-free graphs that do not contain a subdivision of the complete graph on four vertices as an induced subgraph. We prove that every graph of girth at least five in this class is 3-colorable

The world of hereditary graph classes viewed through Truemper configurations

In 1982 Truemper gave a theorem that characterizes graphs whose edges can be labeled so that all chordless cycles have prescribed parities. The characterization states that this can be done for a graph G if and only if it can be done for all induced subgraphs of G that are of few speci c types, that we will call Truemper con gurations. Truemper was originally motivated by the problem of obtaining a co-NP characterization of bipartite graphs that are signable to be balanced (i.e. bipartite graphs whose node-node incidence matrices are balanceable matrices). The con gurations that Truemper identi ed in his theorem ended up playing a key role in understanding the structure of several seemingly diverse classes of objects, such as regular matroids, balanceable matrices and perfect graphs. In this survey we view all these classes, and more, through the excluded Truemper con gurations, focusing on the algorithmic consequences, trying to understand what structurally enables e cient recognition and optimization algorithms

Entire choosability of near-outerplane graphs

It is proved that if G is a plane embedding of a K4-minor-free graph with maximum degree Δ, then G is entirely 7-choosable if Δ≤4 and G is entirely (Δ+ 2)-choosable if Δ≥ 5; that is, if every vertex, edge and face of G is given a list of max{7,Δ+2} colours, then every element can be given a colour from its list such that no two adjacent or incident elements are given the same colour. It is proved also that this result holds if G is a plane embedding of a K2,3-minor-free graph or a (K2 + (K1 U K2))-minor-free graph. As a special case this proves that the Entire Colouring Conjecture, that a plane graph is entirely (Δ + 4)-colourable, holds if G is a plane embedding of a K4-minor-free graph, a K2,3-minor-free graph or a (K2 + (K1 U K2))-minor-free graph