Reconstruction and Edge Reconstruction of Triangle-free Graphs

The Reconstruction Conjecture due to Kelly and Ulam states that every graph with at least 3 vertices is uniquely determined by its multiset of subgraphs $\{G-v: v\in V(G)\}$. Let $diam(G)$ and $\kappa(G)$ denote the diameter and the connectivity of a graph $G$, respectively, and let $\mathcal{G}_2:=\{G: \textrm{diam}(G)=2\}$ and $\mathcal{G}_3:=\{G:\textrm{diam}(G)=\textrm{diam}(\overline{G})=3\}$. It is known that the Reconstruction Conjecture is true if and only if it is true for every 2-connected graph in $\mathcal{G}_2\cup \mathcal{G}_3$. Balakumar and Monikandan showed that the Reconstruction Conjecture holds for every triangle-free graph $G$ in $\mathcal{G}_2\cup \mathcal{G}_3$ with $\kappa(G)=2$. Moreover, they asked whether the result still holds if $\kappa(G)\ge 3$. (If yes, the class of graphs critical for solving the Reconstruction Conjecture is restricted to 2-connected graphs in $\mathcal{G}_2\cup\mathcal{G}_3$ which contain triangles.) In this paper, we give a partial solution to their question by showing that the Reconstruction Conjecture holds for every triangle-free graph $G$ in $\mathcal{G}_3$ and every triangle-free graph $G$ in $\mathcal{G}_2$ with $\kappa(G)=3$. We also prove similar results about the Edge Reconstruction Conjecture.Comment: 11 pages, 3 figure

On the most wanted Folkman graph

We discuss a branch of Ramsey theory concerning edge Folkman numbers. Fe(3; 3; 4) involves the smallest parameters for which the problem is open, posing the question \What is the smallest order N of a K4-free graph, for which any 2-coloring of its edges must contain at least one monochromatic triangle? This is equivalent to finding the order N of the smallest K4-free graph which is not a union of two triangle-free graphs. It is known that 16 is less than or equal to N (an easy bound), and it is known through a probabilistic proof by Spencer that N is less than or equal to 3 X 10^9. In this paper, after overviewing related Folkman problems, we prove that 19 is less than or equal to N, and give some evidence for the bound N is less than equal to 27

Eternal Independent Sets in Graphs

The use of mobile guards to protect a graph has received much attention in the literature of late in the form of eternal dominating sets, eternal vertex covers and other models of graph protection. In this paper, eternal independent sets are introduced. These are independent sets such that the following can be iterated forever: a vertex in the independent set can be replaced with a neighboring vertex and the resulting set is independent

Snarks with total chromatic number 5

A k-total-coloring of G is an assignment of k colors to the edges and vertices of G, so that adjacent and incident elements have different colors. The total chromatic number of G, denoted by chi(T)(G), is the least k for which G has a k-total-coloring. It was proved by Rosenfeld that the total chromatic number of a cubic graph is either 4 or 5. Cubic graphs with chi(T) = 4 are said to be Type 1, and cubic graphs with chi(T) = 5 are said to be Type 2. Snarks are cyclically 4-edge-connected cubic graphs that do not allow a 3-edge-coloring. In 2003, Cavicchioli et al. asked for a Type 2 snark with girth at least 5. As neither Type 2 cubic graphs with girth at least 5 nor Type 2 snarks are known, this is taking two steps at once, and the two requirements of being a snark and having girth at least 5 should better be treated independently. In this paper we will show that the property of being a snark can be combined with being Type 2. We will give a construction that gives Type 2 snarks for each even vertex number n >= 40. We will also give the result of a computer search showing that among all Type 2 cubic graphs on up to 32 vertices, all but three contain an induced chordless cycle of length 4. These three exceptions contain triangles. The question of the existence of a Type 2 cubic graph with girth at least 5 remains open

Exact and Parameterized Algorithms for the Independent Cutset Problem

The Independent Cutset problem asks whether there is a set of vertices in a given graph that is both independent and a cutset. Such a problem is $\textsf{NP}$-complete even when the input graph is planar and has maximum degree five. In this paper, we first present a $\mathcal{O}^*(1.4423^{n})$-time algorithm for the problem. We also show how to compute a minimum independent cutset (if any) in the same running time. Since the property of having an independent cutset is MSO$_1$-expressible, our main results are concerned with structural parameterizations for the problem considering parameters that are not bounded by a function of the clique-width of the input. We present $\textsf{FPT}$-time algorithms for the problem considering the following parameters: the dual of the maximum degree, the dual of the solution size, the size of a dominating set (where a dominating set is given as an additional input), the size of an odd cycle transversal, the distance to chordal graphs, and the distance to $P_5$-free graphs. We close by introducing the notion of $\alpha$-domination, which allows us to identify more fixed-parameter tractable and polynomial-time solvable cases.Comment: 20 pages with references and appendi

Extending a perfect matching to a Hamiltonian cycle

Graph TheoryInternational audienceRuskey and Savage conjectured that in the d-dimensional hypercube, every matching M can be extended to a Hamiltonian cycle. Fink verified this for every perfect matching M, remarkably even if M contains external edges. We prove that this property also holds for sparse spanning regular subgraphs of the cubes: for every d ≥7 and every k, where 7 ≤k ≤d, the d-dimensional hypercube contains a k-regular spanning subgraph such that every perfect matching (possibly with external edges) can be extended to a Hamiltonian cycle. We do not know if this result can be extended to k=4,5,6. It cannot be extended to k=3. Indeed, there are only three 3-regular graphs such that every perfect matching (possibly with external edges) can be extended to a Hamiltonian cycle, namely the complete graph on 4 vertices, the complete bipartite 3-regular graph on 6 vertices and the 3-cube on 8 vertices. Also, we do not know if there are graphs of girth at least 5 with this matching-extendability property

Sangaku in Multiple Geometries: Examining Japanese Temple Geometry Beyond Euclid

When the country of Japan was closed from the rest of the world from 1603 until1867 during the Edo period, the field of mathematics developed in a different wayfrom how it developed in the rest of the world. One way we see this developmentis through the sangaku, the thousands of geometric problems hung in various Shinto and Buddhist temples throughout the country. Written on wooden tablets by people from numerous walks of life, all these problems hold true within Euclidean geometry. During the 1800s, while Japan was still closed, non-Euclidean geometries began to develop across the globe, so the isolated nation was entirely unaware of these new systems. Thus, we will explore the sangaku in two of the other well-known systems, namely the neutral and hyperbolic geometric systems. Specifically, we will highlight how these traditionally-solved problems change under the varying definitions of line parallelism