Results 51 to 60 of about 31,664 (169)
Extremal Graphs for Intersecting Triangles
Journal of Combinatorial Theory, Series B, 1995 A \(k\)-fan is a graph with \(2k+ 1\) vertices consisting of \(k\) 3-cycles having one vertex in common. The authors show that if \(n\geq 50k^2\) and the graph \(G_n\) has more than \([n^2/4]+ k^2- ck\) edges, where \(c\) equals 1 or 3/2 according as \(k\) is odd or even, then \(G_n\) contains a \(k\)-fan; furthermore, the bound for the number of edges
Paul Erdös +3 more
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A characterization of chordal graph without sun and co-rising sun as convex geometry
Theory and Applications of GraphsIn this paper we introduce the notion of $t_3$ \textit{convexity}, a natural restriction of triangle convexity. A \textit{triangle path} is a path allowing just short chords. A triangle path $P$ between two non-adjacent vertices in a graph $G$ is called $
Silvia B. Tondato
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Triangles in randomly perturbed graphs
Combinatorics, Probability and Computing, 2022 AbstractWe study the problem of finding pairwise vertex-disjoint triangles in the randomly perturbed graph model, which is the union of any $n$ -vertex graph $G$ satisfying a given minimum degree condition and the binomial random graph $G(n,p)$ .
Julia Böttcher +3 more
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Triangle-Free Planar Graphs and Segment Intersection Graphs
Journal of Graph Algorithms and Applications, 2002 We prove that every triangle-free planar graph is the intersection graph of a set of segments in the plane. Moreover, the segments can be chosen in only three directions (horizontal, vertical and oblique) and in such a way that no two segments cross, i.e.
Natalia de Castro +4 more
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The spectrum problem for digraphs of order 4 and size 5 [PDF]
Opuscula Mathematica, 2018 The paw graph consists of a triangle with a pendant edge attached to one of the three vertices. We obtain a multigraph by adding exactly one repeated edge to the paw. Now, let \(D\) be a directed graph obtained by orientating the edges of that multigraph.
Ryan C. Bunge +5 more
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Discrete Mathematics & Theoretical Computer Science, 2019 Slimness of a graph measures the local deviation of its metric from a tree metric. In a graph $G=(V,E)$, a geodesic triangle $\bigtriangleup(x,y,z)$ with $x, y, z\in V$ is the union $P(x,y) \cup P(x,z) \cup P(y,z)$ of three shortest paths connecting ...
Feodor F. Dragan, Abdulhakeem Mohammed
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Induced cycles in triangle graphs
Discrete Applied Mathematics, 2016 The triangle graph of a graph $G$, denoted by ${\cal T}(G)$, is the graph whose vertices represent the triangles ($K_3$ subgraphs) of $G$, and two vertices of ${\cal T}(G)$ are adjacent if and only if the corresponding triangles share an edge. In this paper, we characterize graphs whose triangle graph is a cycle and then extend the result to obtain a ...
S. Aparna Lakshmanan +2 more
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Mathematics OpenThe cycle graph of a graph G is the graph [Formula: see text] whose vertices are the induced cycles of G and where two vertices are adjacent if and only if they are distinct induced cycles that share a common edge.
Severino V. Gervacio, Yvette F. Lim
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Triangle‐factors in pseudorandom graphs [PDF]
Bulletin of the London Mathematical Society, 2019 We show that if the second eigenvalue $λ$ of a $d$-regular graph $G$ on $n \in 3 \mathbb{Z}$ vertices is at most $\varepsilon d^2/(n \log n)$, for a small constant $\varepsilon > 0$, then $G$ contains a triangle-factor. The bound on $λ$ is at most an $O(\log n)$ factor away from the best possible one: Krivelevich, Sudakov and Szabó, extending a ...
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Random triangles in random graphs
Random Structures & Algorithms, 2021 AbstractIn a recent paper, Oliver Riordan shows that for and p up to and slightly larger than the threshold for a Kr‐factor, the hypergraph formed by the copies of Kr in G(n, p) contains a copy of the binomial random hypergraph with . For r = 3, he gives a slightly weaker result where the density in the random hypergraph is reduced by a constant ...
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