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The minimum number of vertices for a triangle-free graph with χl(G)=4 is 11

Discrete Mathematics, 2008
It is well-known that the minimum number of vertices for a triangle-free 4-chromatic graph is 11, and the Grötzsch graph is just such a graph. In this paper, we show that every non-bipartite triangle-free graph G of order not greater than 10 has χl(G)=3.
Baoyindureng Wu
exaly   +2 more sources

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Triangles in an Ordinary Graph

Canadian Journal of Mathematics, 1963
An ordinary graph is a finite linear graph which contains no loops or multiple edges, and in which all edges are undirected. In such a graph G, let N, L, and T denote respectively the number of nodes, edges, and triangles. One problem, suggested by P.
Nordhaus, E. A., Stewart, B. M.
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Median Graphs and Triangle-Free Graphs

SIAM Journal on Discrete Mathematics, 1999
Summary: Let \(M(m,n)\) be the complexity of checking whether a graph \(G\) with medges and \(n\) vertices is a median graph. We show that the complexity of checking whether \(G\) is triangle-free is at most \(O(M(m,m))\). Conversely, we prove that the complexity of checking whether a given graph is a median graph is at most \(O(m \log n + T(m \log n,n)
Wilfried Imrich, Sandi Klavzar, Henry Martyn Mulder   +2 more
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The minimum semidefinite rank of a triangle-free graph

Linear Algebra and Its Applications, 2011
We employ a result of Moshe Rosenfeld to show that the minimum semidefinite rank of a triangle-free graph with no isolated vertex must be at least half the number of its vertices.
Louis Deaett
exaly   +2 more sources

Partitioning a triangle-free planar graph into a forest and a forest of bounded degree

Electronic Notes in Discrete Mathematics, 2015
International audienceWe prove that every triangle-free planar graph can have its set of vertices partitioned into two sets, one inducing a forest and the other a forest with maximum degree at most 5.
Mickael Montassier, Alexandre Pinlou
exaly   +2 more sources

CYCLES IN TRIANGLE-FREE GRAPHS

Discrete Mathematics, Algorithms and Applications, 2011
Let G be a k-connected (k ≥ 3), triangle-free graph with α(G) ≤ k + 1. If G is not Petersen graph and G ∉ {Kk, k, Kk, k + 1, Kk + 1, k+1}, then G contains cycles of lengths from 4 to |V(G)|. This generalizes a result conjectured by Amar et al. (Graphs Combin.7 (1991)) and proved by Lou (Discrete Math.152 (1996)).
Xiaojuan Li, Bing Wei 0001, Yongjin Zhu
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A conjecture on triangles of graphs

Graphs and Combinatorics, 1990
The author conjectured in 1981: If a grah G does not contain more than k pairwise edge-disjoint triangles, then there exists a set of at most 2k edges that meets all triangles of G. In the paper this conjecture is proved for various classes of graphs (planar graphs, graphs with n vertices and at least \((7/16)n^ 2\) edges, chordal graphs without a ...
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Triangle graphs

Applied Numerical Mathematics, 1995
The authors introduce a kind of planar graphs which are called triangle graphs and present a way to construct and characterize them. Some properties and applications particularly to the parallel finite element solution of elliptic partial differential equations on triangulated domains are discussed.
Benantar, Messaoud, Doḡrusöz, Uḡur, Flaherty, Joseph E., Krishnamoorthy, Mukkai S.   +3 more
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Realizability of graphs as triangle cover contact graphs

Theoretical Computer Science, 2016
zbMATH Open Web Interface contents unavailable due to conflicting licenses.
Shaheena Sultana, Md. Saidur Rahman 0001
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Triangle graphs and their coloring

2005
In this paper, we present results on two subclasses of trapezoid graphs, including simple trapezoid graphs and triangle graphs (also known as PI graph in [3]). Simple trapezoid graphs and triangle graphs are proper subclasses of trapezoid graphs [5, 3].
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