Displaying similar documents to “Triangle-free planar graphs with minimum degree 3 have radius at least 3”

Note on partitions of planar graphs

Izak Broere, Bonita S. Wilson, Jozef Bucko (2005)

Discussiones Mathematicae Graph Theory

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Chartrand and Kronk in 1969 showed that there are planar graphs whose vertices cannot be partitioned into two parts inducing acyclic subgraphs. In this note we show that the same is true even in the case when one of the partition classes is required to be triangle-free only.

Minimal claw-free graphs

P. Dankelmann, Henda C. Swart, P. van den Berg, Wayne Goddard, M. D. Plummer (2008)

Czechoslovak Mathematical Journal

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A graph G is a minimal claw-free graph (m.c.f. graph) if it contains no K 1 , 3 (claw) as an induced subgraph and if, for each edge e of G , G - e contains an induced claw. We investigate properties of m.c.f. graphs, establish sharp bounds on their orders and the degrees of their vertices, and characterize graphs which have m.c.f. line graphs.

Decompositions of quadrangle-free planar graphs

Oleg V. Borodin, Anna O. Ivanova, Alexandr V. Kostochka, Naeem N. Sheikh (2009)

Discussiones Mathematicae Graph Theory

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W. He et al. showed that a planar graph not containing 4-cycles can be decomposed into a forest and a graph with maximum degree at most 7. This degree restriction was improved to 6 by Borodin et al. We further lower this bound to 5 and show that it cannot be improved to 3.

Cycles through specified vertices in triangle-free graphs

Daniel Paulusma, Kiyoshi Yoshimoto (2007)

Discussiones Mathematicae Graph Theory

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Let G be a triangle-free graph with δ(G) ≥ 2 and σ₄(G) ≥ |V(G)| + 2. Let S ⊂ V(G) consist of less than σ₄/4+ 1 vertices. We prove the following. If all vertices of S have degree at least three, then there exists a cycle C containing S. Both the upper bound on |S| and the lower bound on σ₄ are best possible.

On the Non-(p−1)-Partite Kp-Free Graphs

Kinnari Amin, Jill Faudree, Ronald J. Gould, Elżbieta Sidorowicz (2013)

Discussiones Mathematicae Graph Theory

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We say that a graph G is maximal Kp-free if G does not contain Kp but if we add any new edge e ∈ E(G) to G, then the graph G + e contains Kp. We study the minimum and maximum size of non-(p − 1)-partite maximal Kp-free graphs with n vertices. We also answer the interpolation question: for which values of n and m are there any n-vertex maximal Kp-free graphs of size m?

Paired- and induced paired-domination in {E,net}-free graphs

Oliver Schaudt (2012)

Discussiones Mathematicae Graph Theory

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A dominating set of a graph is a vertex subset that any vertex belongs to or is adjacent to. Among the many well-studied variants of domination are the so-called paired-dominating sets. A paired-dominating set is a dominating set whose induced subgraph has a perfect matching. In this paper, we continue their study. We focus on graphs that do not contain the net-graph (obtained by attaching a pendant vertex to each vertex of the triangle) or the E-graph (obtained by...