Independence Number in Path Graphs

In the paper we present results, which allow us to compute the independence numbers of $P_2$-path graphs and $P_3$-path graphs of special graphs. As $P_2(G)$ and $P_3(G)$ are subgraphs of iterated line graphs $L^2(G)$ and $L^3(G)$, respectively, we compare our results with the independence numbers of corresponding iterated line graphs

Hamiltonian path and Hamiltonian cycle are solvable in polynomial time in graphs of bounded independence number

A Hamiltonian path (a Hamiltonian cycle) in a graph is a path (a cycle, respectively) that traverses all of its vertices. The problems of deciding their existence in an input graph are well-known to be NP-complete, in fact, they belong to the first problems shown to be computationally hard when the theory of NP-completeness was being developed. A lot of research has been devoted to the complexity of Hamiltonian path and Hamiltonian cycle problems for special graph classes, yet only a handful of positive results are known. The complexities of both of these problems have been open even for $4K_1$-free graphs, i.e., graphs of independence number at most $3$. We answer this question in the general setting of graphs of bounded independence number. We also consider a newly introduced problem called \emph{Hamiltonian-$\ell$-Linkage} which is related to the notions of a path cover and of a linkage in a graph. This problem asks if given $\ell$ pairs of vertices in an input graph can be connected by disjoint paths that altogether traverse all vertices of the graph. For $\ell=1$, Hamiltonian-1-Linkage asks for existence of a Hamiltonian path connecting a given pair of vertices. Our main result reads that for every pair of integers $k$ and $\ell$, the Hamiltonian-$\ell$-Linkage problem is polynomial time solvable for graphs of independence number not exceeding $k$. We further complement this general polynomial time algorithm by a structural description of obstacles to Hamiltonicity in graphs of independence number at most $k$ for small values of $k$

Paths, cycles and wheels in graphs without antitriangles

We investigate paths, cycles and wheels in graphs with independence number of at most 2, in particular we prove theorems characterizing all such graphs which are hamiltonian. Ramsey numbers of the form R (G,K3), for G being a path, a cycle or a wheel, are known to be 2n (G) - 1, except for some small cases. In this paper we derive and count all critical graphs 1 for these Ramsey numbers

On monophonic position sets in graphs

The general position problem in graph theory asks for the largest set S of vertices of a graph G such that no shortest path of G contains more than two vertices of S. In this paper we consider a variant of the general position problem called the monophonic position problem, obtained by replacing ‘shortest path’ by ‘induced path’. We prove some basic properties and bounds for the monophonic position number of a graph and determine the monophonic position number of some graph families, including unicyclic graphs, complements of bipartite graphs and split graphs. We show that the monophonic position number of triangle-free graphs is bounded above by the independence number. We present realisation results for the general position number, monophonic position number and monophonic hull number. Finally we discuss the complexity of the monophonic position problem

Coloring Graphs with Forbidden Minors

Hadwiger's conjecture from 1943 states that for every integer $t\ge1$, every graph either can be $t$-colored or has a subgraph that can be contracted to the complete graph on $t+1$ vertices. As pointed out by Paul Seymour in his recent survey on Hadwiger's conjecture, proving that graphs with no $K_7$ minor are $6$-colorable is the first case of Hadwiger's conjecture that is still open. It is not known yet whether graphs with no $K_7$ minor are $7$-colorable. Using a Kempe-chain argument along with the fact that an induced path on three vertices is dominating in a graph with independence number two, we first give a very short and computer-free proof of a recent result of Albar and Gon\c{c}alves and generalize it to the next step by showing that every graph with no $K_t$ minor is $(2t-6)$-colorable, where $t\in\{7,8,9\}$. We then prove that graphs with no $K_8^-$ minor are $9$-colorable and graphs with no $K_8^=$ minor are $8$-colorable. Finally we prove that if Mader's bound for the extremal function for $K_p$ minors is true, then every graph with no $K_p$ minor is $(2t-6)$-colorable for all $p\ge5$. This implies our first result. We believe that the Kempe-chain method we have developed in this paper is of independent interest