Hamilton cycles in almost distance-hereditary graphs

Let $G$ be a graph on $n\geq 3$ vertices. A graph $G$ is almost distance-hereditary if each connected induced subgraph $H$ of $G$ has the property $d_{H}(x,y)\leq d_{G}(x,y)+1$ for any pair of vertices $x,y\in V(H)$. A graph $G$ is called 1-heavy (2-heavy) if at least one (two) of the end vertices of each induced subgraph of $G$ isomorphic to $K_{1,3}$ (a claw) has (have) degree at least $n/2$, and called claw-heavy if each claw of $G$ has a pair of end vertices with degree sum at least $n$. Thus every 2-heavy graph is claw-heavy. In this paper we prove the following two results: (1) Every 2-connected, claw-heavy and almost distance-hereditary graph is Hamiltonian. (2) Every 3-connected, 1-heavy and almost distance-hereditary graph is Hamiltonian. In particular, the first result improves a previous theorem of Feng and Guo. Both results are sharp in some sense.Comment: 14 pages; 1 figure; a new theorem is adde

Sets as graphs

The aim of this thesis is a mutual transfer of computational and structural results and techniques between sets and graphs. We study combinatorial enumeration of sets, canonical encodings, random generation, digraph immersions. We also investigate the underlying structure of sets in algorithmic terms, or in connection with hereditary graphs classes. Finally, we employ a set-based proof-checker to verify two classical results on claw-free graph

Pancyclicity of highly connected graphs

A well-known result due to Chvat\'al and Erd\H{o}s (1972) asserts that, if a graph $G$ satisfies $\kappa(G) \ge \alpha(G)$, where $\kappa(G)$ is the vertex-connectivity of $G$, then $G$ has a Hamilton cycle. We prove a similar result implying that a graph $G$ is pancyclic, namely it contains cycles of all lengths between $3$ and $|G|$: if $|G|$ is large and $\kappa(G) > \alpha(G)$, then $G$ is pancyclic. This confirms a conjecture of Jackson and Ordaz (1990) for large graphs, and improves upon a very recent result of Dragani\'c, Munh\'a-Correia, and Sudakov.Comment: 31 pages, 11 figure

Twin-constrained Hamiltonian paths on threshold graphs: an approach to the minimum score separation problem

The Minimum Score Separation Problem (MSSP) is a combinatorial problem that has been introduced in JORS 55 as an open problem in the paper industry arising in conjunction with the cutting-stock problem. During the process of producing boxes, áat papers are prepared for folding by being scored with knives. The problem is to determine if and how a given production pattern of boxes can be arranged such that a certain minimum distance between the knives can be kept. While it was originally suggested to analyse the MSSP as a specific variant of a Generalized Travelling Salesman Problem, the thesis introduces the concept of twin-constrained Hamiltonian cycles and models the MSSP as the problem of finding a twin-constrained Hamiltonian path on a threshold graph (threshold graphs are a specific type of interval graphs). For a given undirected graph G(N,E) with an even node set N and edge set E, and a bijective function b on N that assigns to every node i in N a "twin node" b(i)6=i, we define a new graph G'(N,E') by adding the edges {i,b(i)} to E. The graph G is said to have a twin-constrained Hamiltonian path with respect to b if there exists a Hamiltonian path on G' in which every node has its twin node as its predecessor (or successor). We start with presenting some general Öndings for the construction of matchings, alternating paths, Hamiltonian paths and alternating cycles on threshold graphs. On this basis it is possible to develop criteria that allow for the construction of twin-constrained Hamiltonian paths on threshold graphs and lead to a heuristic that can quickly solve a large percentage of instances of the MSSP. The insights gained in this way can be generalized and lead to an (exact) polynomial time algorithm for the MSSP. Computational experiments for both the heuristic and the polynomial-time algorithm demonstrate the efficiency of our approach to the MSSP. Finally, possible extensions of the approach are presented

Extremal problems on cycle structure and colorings of graphs

In this Thesis, we consider two main themes: conditions that guarantee diverse cycle structure within a graph, and the existence of strong edge-colorings for a specific family of graphs. In Chapter 2 we consider a question closely related to the Matthews-Sumner conjecture, which states that every 4-connected claw-free graph is Hamiltonian. Since there exists an infinite family of 4-connected claw-free graphs that are not pancyclic, Gould posed the problem of characterizing the pairs of graphs, {X,Y}, such that every 4-connected {X,Y}-free graph is pancyclic. In this chapter we describe a family of pairs of graphs such that if every 4-connected {X,Y}-free graph is pancyclic, then {X,Y} is in this family. Furthermore, we show that every 4-connected {K_(1,3),N(4,1,1)}-free graph is pancyclic. This result, together with several others, completes a characterization of the family of subgraphs, F such that for all H in ∈, every 4-connected {K_(1,3), H}-free graph is pancyclic. In Chapters and 4 we consider refinements of results on cycles and chorded cycles. In 1963, Corrádi and Hajnal proved a conjecture of Erdös, showing that every graph G on at least 3k vertices with minimum degree at least 2k contains k disjoint cycles. This result was extended by Enomoto and Wang, who independently proved that graphs on at least 3kvertices with minimum degree-sum at least 4k - 1 also contain k disjoint cycles. Both results are best possible, and recently, Kierstead, Kostochka, Molla, and Yeager characterized their sharpness examples. A chorded cycle analogue to the result of Corrádi and Hajnal was proved by Finkel, and a similar analogue to the result of Enomoto and Wang was proved by Chiba, Fujita, Gao, and Li. In Chapter 3 we characterize the sharpness examples to these statements, which provides a chorded cycle analogue to the characterization of Kierstead et al. In Chapter 4 we consider another result of Chiba et al., which states that for all integers r and s with r + s ≥ 1, every graph G on at least 3r + 4s vertices with ẟ(G) ≥ 2r+3s contains r disjoint cycles and s disjoint chorded cycles. We provide a characterization of the sharpness examples to this result, which yields a transition between the characterization of Kierstead et al. and the main result of Chapter 3. In Chapter 5 we move to the topic of edge-colorings, considering a variation known as strong edge-coloring. In 1990, Faudree, Gyárfás, Schelp, and Tuza posed several conjectures regarding strong edge-colorings of subcubic graphs. In particular, they conjectured that every subcubic planar graph has a strong edge-coloring using at most nine colors. We prove a slightly stronger form of this conjecture, showing that it holds for all subcubic planar loopless multigraphs

Exploiting structure to cope with NP-hard graph problems: Polynomial and exponential time exact algorithms

An ideal algorithm for solving a particular problem always finds an optimal solution, finds such a solution for every possible instance, and finds it in polynomial time. When dealing with NP-hard problems, algorithms can only be expected to possess at most two out of these three desirable properties. All algorithms presented in this thesis are exact algorithms, which means that they always find an optimal solution. Demanding the solution to be optimal means that other concessions have to be made when designing an exact algorithm for an NP-hard problem: we either have to impose restrictions on the instances of the problem in order to achieve a polynomial time complexity, or we have to abandon the requirement that the worst-case running time has to be polynomial. In some cases, when the problem under consideration remains NP-hard on restricted input, we are even forced to do both. Most of the problems studied in this thesis deal with partitioning the vertex set of a given graph. In the other problems the task is to find certain types of paths and cycles in graphs. The problems all have in common that they are NP-hard on general graphs. We present several polynomial time algorithms for solving restrictions of these problems to specific graph classes, in particular graphs without long induced paths, chordal graphs and claw-free graphs. For problems that remain NP-hard even on restricted input we present exact exponential time algorithms. In the design of each of our algorithms, structural graph properties have been heavily exploited. Apart from using existing structural results, we prove new structural properties of certain types of graphs in order to obtain our algorithmic results