14,868 research outputs found
Fractional coloring of triangle-free planar graphs
We prove that every planar triangle-free graph on vertices has fractional
chromatic number at most
Vertex arboricity of triangle-free graphs
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
On the size of identifying codes in triangle-free graphs
In an undirected graph , a subset such that is a
dominating set of , and each vertex in is dominated by a distinct
subset of vertices from , is called an identifying code of . The concept
of identifying codes was introduced by Karpovsky, Chakrabarty and Levitin in
1998. For a given identifiable graph , let \M(G) be the minimum
cardinality of an identifying code in . In this paper, we show that for any
connected identifiable triangle-free graph on vertices having maximum
degree , \M(G)\le n-\tfrac{n}{\Delta+o(\Delta)}. This bound is
asymptotically tight up to constants due to various classes of graphs including
-ary trees, which are known to have their minimum identifying code
of size . We also provide improved bounds for
restricted subfamilies of triangle-free graphs, and conjecture that there
exists some constant such that the bound \M(G)\le n-\tfrac{n}{\Delta}+c
holds for any nontrivial connected identifiable graph
Applications of a new separator theorem for string graphs
An intersection graph of curves in the plane is called a string graph.
Matousek almost completely settled a conjecture of the authors by showing that
every string graph of m edges admits a vertex separator of size O(\sqrt{m}\log
m). In the present note, this bound is combined with a result of the authors,
according to which every dense string graph contains a large complete balanced
bipartite graph. Three applications are given concerning string graphs G with n
vertices: (i) if K_t is not a subgraph of G for some t, then the chromatic
number of G is at most (\log n)^{O(\log t)}; (ii) if K_{t,t} is not a subgraph
of G, then G has at most t(\log t)^{O(1)}n edges,; and (iii) a lopsided
Ramsey-type result, which shows that the Erdos-Hajnal conjecture almost holds
for string graphs.Comment: 7 page
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