688 research outputs found

    Vertex arboricity of triangle-free graphs

    Get PDF
    Master's Project (M.S.) University of Alaska Fairbanks, 2016The vertex arboricity of a graph is the minimum number of colors needed to color the vertices so that the subgraph induced by each color class is a forest. In other words, the vertex arboricity of a graph is the fewest number of colors required in order to color a graph such that every cycle has at least two colors. Although not standard, we will refer to vertex arboricity simply as arboricity. In this paper, we discuss properties of chromatic number and k-defective chromatic number and how those properties relate to the arboricity of trianglefree graphs. In particular, we find bounds on the minimum order of a graph having arboricity three. Equivalently, we consider the largest possible vertex arboricity of triangle-free graphs of fixed order

    Automated Discharging Arguments for Density Problems in Grids

    Full text link
    Discharging arguments demonstrate a connection between local structure and global averages. This makes it an effective tool for proving lower bounds on the density of special sets in infinite grids. However, the minimum density of an identifying code in the hexagonal grid remains open, with an upper bound of 37≈0.428571\frac{3}{7} \approx 0.428571 and a lower bound of 512≈0.416666\frac{5}{12}\approx 0.416666. We present a new, experimental framework for producing discharging arguments using an algorithm. This algorithm replaces the lengthy case analysis of human-written discharging arguments with a linear program that produces the best possible lower bound using the specified set of discharging rules. We use this framework to present a lower bound of 2355≈0.418181\frac{23}{55} \approx 0.418181 on the density of an identifying code in the hexagonal grid, and also find several sharp lower bounds for variations on identifying codes in the hexagonal, square, and triangular grids.Comment: This is an extended abstract, with 10 pages, 2 appendices, 5 tables, and 2 figure

    THE ELECTRONIC JOURNAL OF COMBINATORICS (2014), DS1.14 References

    Get PDF
    and Computing 11. The results of 143 references depend on computer algorithms. The references are ordered alphabetically by the last name of the first author, and where multiple papers have the same first author they are ordered by the last name of the second author, etc. We preferred that all work by the same author be in consecutive positions. Unfortunately, this causes that some of the abbreviations are not in alphabetical order. For example, [BaRT] is earlier on the list than [BaLS]. We also wish to explain a possible confusion with respect to the order of parts and spelling of Chinese names. We put them without any abbreviations, often with the last name written first as is customary in original. Sometimes this is different from the citations in other sources. One can obtain all variations of writing any specific name by consulting the authors database of Mathematical Reviews a

    The zero forcing polynomial of a graph

    Full text link
    Zero forcing is an iterative graph coloring process, where given a set of initially colored vertices, a colored vertex with a single uncolored neighbor causes that neighbor to become colored. A zero forcing set is a set of initially colored vertices which causes the entire graph to eventually become colored. In this paper, we study the counting problem associated with zero forcing. We introduce the zero forcing polynomial of a graph GG of order nn as the polynomial Z(G;x)=∑i=1nz(G;i)xi\mathcal{Z}(G;x)=\sum_{i=1}^n z(G;i) x^i, where z(G;i)z(G;i) is the number of zero forcing sets of GG of size ii. We characterize the extremal coefficients of Z(G;x)\mathcal{Z}(G;x), derive closed form expressions for the zero forcing polynomials of several families of graphs, and explore various structural properties of Z(G;x)\mathcal{Z}(G;x), including multiplicativity, unimodality, and uniqueness.Comment: 23 page

    New constructions for covering designs

    Full text link
    A (v,k,t)(v,k,t) {\em covering design}, or {\em covering}, is a family of kk-subsets, called blocks, chosen from a vv-set, such that each tt-subset is contained in at least one of the blocks. The number of blocks is the covering's {\em size}, and the minimum size of such a covering is denoted by C(v,k,t)C(v,k,t). This paper gives three new methods for constructing good coverings: a greedy algorithm similar to Conway and Sloane's algorithm for lexicographic codes~\cite{lex}, and two methods that synthesize new coverings from preexisting ones. Using these new methods, together with results in the literature, we build tables of upper bounds on C(v,k,t)C(v,k,t) for v≤32v \leq 32, k≤16k \leq 16, and t≤8t \leq 8.
    • …
    corecore