Rainbow Connection Number and Connected Dominating Sets

Rainbow connection number rc(G) of a connected graph G is the minimum number of colours needed to colour the edges of G, so that every pair of vertices is connected by at least one path in which no two edges are coloured the same. In this paper we show that for every connected graph G, with minimum degree at least 2, the rainbow connection number is upper bounded by {\gamma}_c(G) + 2, where {\gamma}_c(G) is the connected domination number of G. Bounds of the form diameter(G) \leq rc(G) \leq diameter(G) + c, 1 \leq c \leq 4, for many special graph classes follow as easy corollaries from this result. This includes interval graphs, AT-free graphs, circular arc graphs, threshold graphs, and chain graphs all with minimum degree at least 2 and connected. We also show that every bridge-less chordal graph G has rc(G) \leq 3.radius(G). In most of these cases, we also demonstrate the tightness of the bounds. An extension of this idea to two-step dominating sets is used to show that for every connected graph on n vertices with minimum degree {\delta}, the rainbow connection number is upper bounded by 3n/({\delta} + 1) + 3. This solves an open problem of Schiermeyer (2009), improving the previously best known bound of 20n/{\delta} by Krivelevich and Yuster (2010). Moreover, this bound is seen to be tight up to additive factors by a construction of Caro et al. (2008).Comment: 14 page

The number and average size of connected sets in graphs with degree constraints

The average size of connected vertex subsets of a connected graph generalises a much‐studied parameter for subtrees of trees. For trees, the possible values of this parameter are critically affected by the presence or absence of vertices of degree 2. We answer two questions of Andrew Vince regarding the effect of degree constraints on general connected graphs. We give a new lower bound, and the first nontrivial upper bound, on the maximum growth rate of the number of connected sets of a cubic graph, and in fact obtain nontrivial upper bounds for any constant bound on the maximum degree. We show that the average connected set density is bounded away from 1 for graphs with no vertex of degree 2, and generalise a classical result of Jamison for trees by showing that in order for the connected set density to approach 1, the proportion of vertices of degree 2 must approach 1. Finally, we show that any sequence of graphs with minimum degree tending to infinity must have connected set density tending to 1/2

Reduction Techniques for Graph Isomorphism in the Context of Width Parameters

We study the parameterized complexity of the graph isomorphism problem when parameterized by width parameters related to tree decompositions. We apply the following technique to obtain fixed-parameter tractability for such parameters. We first compute an isomorphism invariant set of potential bags for a decomposition and then apply a restricted version of the Weisfeiler-Lehman algorithm to solve isomorphism. With this we show fixed-parameter tractability for several parameters and provide a unified explanation for various isomorphism results concerned with parameters related to tree decompositions. As a possibly first step towards intractability results for parameterized graph isomorphism we develop an fpt Turing-reduction from strong tree width to the a priori unrelated parameter maximum degree.Comment: 23 pages, 4 figure

Extremal problems on special graph colorings

In this thesis, we study several extremal problems on graph colorings. In particular, we study monochromatic connected matchings, paths, and cycles in 2-edge colored graphs, packing colorings of subcubic graphs, and directed intersection number of digraphs. In Chapter 2, we consider monochromatic structures in 2-edge colored graphs. A matching M in a graph G is connected if all the edges of M are in the same component of G. Following Łuczak, there are a number of results using the existence of large connected matchings in cluster graphs with respect to regular partitions of large graphs to show the existence of long paths and other structures in these graphs. We prove exact Ramsey-type bounds on the sizes of monochromatic connected matchings in 2-edge-colored multipartite graphs. In addition, we prove a stability theorem for such matchings, which is used to find necessary and sufficient conditions on the existence of monochromatic paths and cycles: for every fixed s and large n, we describe all values of n_1, ...,n_s such that for every 2-edge-coloring of the complete s-partite graph K_{n_1, ...,n_s} there exists a monochromatic (i) cycle C_{2n} with 2n vertices, (ii) cycle C_{at least 2n} with at least 2n vertices, (iii) path P_{2n} with 2n vertices, and (iv) path P_{2n+1} with 2n+1 vertices. Our results also imply for large n of the conjecture by Gyárfás, Ruszinkó, Sárkőzy and Szemerédi that for every 2-edge-coloring of the complete 3-partite graph K_{n,n,n} there is a monochromatic path P_{2n+1}. Moreover, we prove that for every sufficiently large n, if n = 3t+r where r in {0,1,2} and G is an n-vertex graph with minimum degree at least (3n-1)/4, then for every 2-edge-coloring of G, either there are cycles of every length {3, 4, 5, ..., 2t+r} of the same color, or there are cycles of every even length {4, 6, 8, ..., 2t+2} of the same color. This result is tight and implies the conjecture of Schelp that for every sufficiently large n, every (3n-1)-vertex graph G with minimum degree larger than 3|V(G)|/4, in each 2-edge-coloring of G there exists a monochromatic path P_{2n} with 2n vertices. It also implies for sufficiently large n the conjecture by Benevides, Łuczak, Scott, Skokan and White that for every positive integer n of the form n=3t+r where r in {0,1,2} and every n-vertex graph G with minimum degree at least 3n/4, in each 2-edge-coloring of G there exists a monochromatic cycle of length at least 2t+r. In Chapter 3, we consider a collection of special vertex colorings called packing colorings. For a sequence of non-decreasing positive integers S = (s_1, ..., s_k), a packing S-coloring is a partition of V(G) into sets V_1, ..., V_k such that for each integer i in {1, ..., k} the distance between any two distinct x,y in V_i is at least s_i+1. The smallest k such that G has a packing (1,2, ..., k)-coloring is called the packing chromatic number of G and is denoted by \chi_p(G). The question whether the packing chromatic number of subcubic graphs is bounded appears in several papers. We show that for every fixed k and g at least 2k+2, almost every n-vertex cubic graph of girth at least g has the packing chromatic number greater than k, which answers the previous question in the negative. Moreover, we work towards the conjecture of Brešar, Klavžar, Rall and Wash that the packing chromatic number of 1-subdivision of subcubic graphs are bounded above by 5. In particular, we show that every subcubic graph is (1,1,2,2,3,3,k)-colorable for every integer k at least 4 via a coloring in which color k is used at most once, every 2-degenerate subcubic graph is (1,1,2,2,3,3)-colorable, and every subcubic graph with maximum average degree less than 30/11 is packing (1,1,2,2)-colorable. Furthermore, while proving the packing chromatic number of subcubic graphs is unbounded, we also consider improving upper bound on the independence ratio, alpha(G)/n, of cubic n-vertex graphs of large girth. We show that ``almost all" cubic labeled graphs of girth at least 16 have independence ratio at most 0.454. In Chapter 4, we introduce and study the directed intersection representation of digraphs. A directed intersection representation is an assignment of a color set to each vertex in a digraph such that two vertices form an edge if and only if their color sets share at least one color and the tail vertex has a strictly smaller color set than the head. The smallest possible size of the union of the color sets is defined to be the directed intersection number (DIN). We show that the directed intersection representation is well-defined for all directed acyclic graphs and the maximum DIN among all n vertex acyclic digraphs is at most 5n^2/8 + O(n) and at least 9n^2/16 + O(n)

Flip Graphs of Degree-Bounded (Pseudo-)Triangulations

We study flip graphs of triangulations whose maximum vertex degree is bounded by a constant $k$. In particular, we consider triangulations of sets of $n$ points in convex position in the plane and prove that their flip graph is connected if and only if $k > 6$; the diameter of the flip graph is $O(n^2)$. We also show that, for general point sets, flip graphs of pointed pseudo-triangulations can be disconnected for $k \leq 9$, and flip graphs of triangulations can be disconnected for any $k$. Additionally, we consider a relaxed version of the original problem. We allow the violation of the degree bound $k$ by a small constant. Any two triangulations with maximum degree at most $k$ of a convex point set are connected in the flip graph by a path of length $O(n \log n)$, where every intermediate triangulation has maximum degree at most $k+4$.Comment: 13 pages, 12 figures, acknowledgments update

Deterministic polynomial-time approximation algorithms for partition functions and graph polynomials

In this paper we show a new way of constructing deterministic polynomial-time approximation algorithms for computing complex-valued evaluations of a large class of graph polynomials on bounded degree graphs. In particular, our approach works for the Tutte polynomial and independence polynomial, as well as partition functions of complex-valued spin and edge-coloring models. More specifically, we define a large class of graph polynomials $\mathcal C$ and show that if $p\in \cal C$ and there is a disk $D$ centered at zero in the complex plane such that $p(G)$ does not vanish on $D$ for all bounded degree graphs $G$, then for each $z$ in the interior of $D$ there exists a deterministic polynomial-time approximation algorithm for evaluating $p(G)$ at $z$. This gives an explicit connection between absence of zeros of graph polynomials and the existence of efficient approximation algorithms, allowing us to show new relationships between well-known conjectures. Our work builds on a recent line of work initiated by. Barvinok, which provides a new algorithmic approach besides the existing Markov chain Monte Carlo method and the correlation decay method for these types of problems.Comment: 27 pages; some changes have been made based on referee comments. In particular a tiny error in Proposition 4.4 has been fixed. The introduction and concluding remarks have also been rewritten to incorporate the most recent developments. Accepted for publication in SIAM Journal on Computatio

Dynamic programming for graphs on surfaces

We provide a framework for the design and analysis of dynamic programming algorithms for surface-embedded graphs on n vertices and branchwidth at most k. Our technique applies to general families of problems where standard dynamic programming runs in 2O(k·log k). Our approach combines tools from topological graph theory and analytic combinatorics.Postprint (updated version

On the stable degree of graphs

We define the stable degree s(G) of a graph G by s(G)∈=∈ min max d (v), where the minimum is taken over all maximal independent sets U of G. For this new parameter we prove the following. Deciding whether a graph has stable degree at most k is NP-complete for every fixed k∈≥∈3; and the stable degree is hard to approximate. For asteroidal triple-free graphs and graphs of bounded asteroidal number the stable degree can be computed in polynomial time. For graphs in these classes the treewidth is bounded from below and above in terms of the stable degree