Complexity of colouring problems restricted to unichord-free and \{square,unichord\}-free graphs

A \emph{unichord} in a graph is an edge that is the unique chord of a cycle. A \emph{square} is an induced cycle on four vertices. A graph is \emph{unichord-free} if none of its edges is a unichord. We give a slight restatement of a known structure theorem for unichord-free graphs and use it to show that, with the only exception of the complete graph $K_4$, every square-free, unichord-free graph of maximum degree~3 can be total-coloured with four colours. Our proof can be turned into a polynomial time algorithm that actually outputs the colouring. This settles the class of square-free, unichord-free graphs as a class for which edge-colouring is NP-complete but total-colouring is polynomial

Graphs that do not contain a cycle with a node that has at least two neighbors on it

We recall several known results about minimally 2-connected graphs, and show that they all follow from a decomposition theorem. Starting from an analogy with critically 2-connected graphs, we give structural characterizations of the classes of graphs that do not contain as a subgraph and as an induced subgraph, a cycle with a node that has at least two neighbors on the cycle. From these characterizations we get polynomial time recognition algorithms for these classes, as well as polynomial time algorithms for vertex-coloring and edge-coloring

Acyclic Chromatic Index of Chordless Graphs

An acyclic edge coloring of a graph is a proper edge coloring in which there are no bichromatic cycles. The acyclic chromatic index of a graph $G$ denoted by $a'(G)$, is the minimum positive integer $k$ such that $G$ has an acyclic edge coloring with $k$ colors. It has been conjectured by Fiam\v{c}\'{\i}k that $a'(G) \le \Delta+2$ for any graph $G$ with maximum degree $\Delta$. Linear arboricity of a graph $G$, denoted by $la(G)$, is the minimum number of linear forests into which the edges of $G$ can be partitioned. A graph is said to be chordless if no cycle in the graph contains a chord. Every $2$-connected chordless graph is a minimally $2$-connected graph. It was shown by Basavaraju and Chandran that if $G$ is $2$-degenerate, then $a'(G) \le \Delta+1$. Since chordless graphs are also $2$-degenerate, we have $a'(G) \le \Delta+1$ for any chordless graph $G$. Machado, de Figueiredo and Trotignon proved that the chromatic index of a chordless graph is $\Delta$ when $\Delta \ge 3$. They also obtained a polynomial time algorithm to color a chordless graph optimally. We improve this result by proving that the acyclic chromatic index of a chordless graph is $\Delta$, except when $\Delta=2$ and the graph has a cycle, in which case it is $\Delta+1$. We also provide the sketch of a polynomial time algorithm for an optimal acyclic edge coloring of a chordless graph. As a byproduct, we also prove that $la(G) = \lceil \frac{\Delta }{2} \rceil$, unless $G$ has a cycle with $\Delta=2$, in which case $la(G) = \lceil \frac{\Delta+1}{2} \rceil = 2$. To obtain the result on acyclic chromatic index, we prove a structural result on chordless graphs which is a refinement of the structure given by Machado, de Figueiredo and Trotignon for this class of graphs. This might be of independent interest

The (theta, wheel)-free graphs Part I: Only-prism and only-pyramid graphs

Truemper configurations are four types of graphs (namely thetas, wheels, prisms and pyramids) that play an important role in the proof of several decomposition theorems for hereditary graph classes. In this paper, we prove two structure theorems: one for graphs with no thetas, wheels and prisms as induced subgraphs, and one for graphs with no thetas, wheels and pyramids as induced subgraphs. A consequence is a polynomial time recognition algorithms for these two classes. In Part II of this series we generalize these results to graphs with no thetas and wheels as induced subgraphs, and in Parts III and IV, using the obtained structure, we solve several optimization problems for these graphs

Proper connection number of graphs

The concept of \emph{proper connection number} of graphs is an extension of proper colouring and is motivated by rainbow connection number of graphs. Let $G$ be an edge-coloured graph. Andrews et al.\cite{Andrews2016} and, independently, Borozan et al.\cite{Borozan2012} introduced the concept of proper connection number as follows: A coloured path $P$ in an edge-coloured graph $G$ is called a \emph{properly coloured path} or more simple \emph{proper path} if two any consecutive edges receive different colours. An edge-coloured graph $G$ is called a \emph{properly connected graph} if every pair of vertices is connected by a proper path. The \emph{proper connection number}, denoted by $pc(G)$, of a connected graph $G$ is the smallest number of colours that are needed in order to make $G$ properly connected. Let $k\geq2$ be an integer. If every two vertices of an edge-coloured graph $G$ are connected by at least $k$ proper paths, then $G$ is said to be a \emph{properly $k$-connected graph}. The \emph{proper $k$-connection number} $pc_k(G)$, introduced by Borozan et al. \cite{Borozan2012}, is the smallest number of colours that are needed in order to make $G$ a properly $k$-connected graph. The aims of this dissertation are to study the proper connection number and the proper 2-connection number of several classes of connected graphs. All the main results are contained in Chapter 4, Chapter 5 and Chapter 6. Since every 2-connected graph has proper connection number at most 3 by Borozan et al. \cite{Borozan2012} and the proper connection number of a connected graph $G$ equals 1 if and only if $G$ is a complete graph by the authors in \cite{Andrews2016, Borozan2012}, our motivation is to characterize 2-connected graphs which have proper connection number 2. First of all, we disprove Conjecture 3 in \cite{Borozan2012} by constructing classes of 2-connected graphs with minimum degree $\delta(G)\geq3$ that have proper connection number 3. Furthermore, we study sufficient conditions in terms of the ratio between the minimum degree and the order of a 2-connected graph $G$ implying that $G$ has proper connection number 2. These results are presented in Chapter 4 of the dissertation. In Chapter 5, we study proper connection number at most 2 of connected graphs in the terms of connectivity and forbidden induced subgraphs $S_{i,j,k}$, where $i,j,k$ are three integers and $0\leq i\leq j\leq k$ (where $S_{i,j,k}$ is the graph consisting of three paths with $i,j$ and $k$ edges having an end-vertex in common). Recently, there are not so many results on the proper $k$-connection number $pc_k(G)$, where $k\geq2$ is an integer. Hence, in Chapter 6, we consider the proper 2-connection number of several classes of connected graphs. We prove a new upper bound for $pc_2(G)$ and determine several classes of connected graphs satisfying $pc_2(G)=2$. Among these are all graphs satisfying the Chv\'{a}tal and Erd\'{o}s condition ($\alpha({G})\leq\kappa(G)$ with two exceptions). We also study the relationship between proper 2-connection number $pc_2(G)$ and proper connection number $pc(G)$ of the Cartesian product of two nontrivial connected graphs. In the last chapter of the dissertation, we propose some open problems of the proper connection number and the proper 2-connection number