Graphs isomorphic to their path graphs

Martin Knor; Ľudovít Niepel

Displaying similar documents to “Graphs isomorphic to their path graphs”

A Fan-Type Heavy Pair Of Subgraphs For Pancyclicity Of 2-Connected Graphs

Wojciech Wideł (2016)

Discussiones Mathematicae Graph Theory

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Let G be a graph on n vertices and let H be a given graph. We say that G is pancyclic, if it contains cycles of all lengths from 3 up to n, and that it is H-f1-heavy, if for every induced subgraph K of G isomorphic to H and every two vertices u, v ∈ V (K), dK(u, v) = 2 implies [...] min⁡dG(u),dG(v)≥n+12 $min {d_{G} (u), d_{G} (v)} \geq \frac{n + 1}{2}$ . In this paper we prove that every 2-connected K1,3, P5-f1-heavy graph is pancyclic. This result completes the answer to the problem of finding f1-heavy pairs of subgraphs implying...

The structure and existence of 2-factors in iterated line graphs

Michael Ferrara, Ronald J. Gould, Stephen G. Hartke (2007)

Discussiones Mathematicae Graph Theory

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We prove several results about the structure of 2-factors in iterated line graphs. Specifically, we give degree conditions on G that ensure L²(G) contains a 2-factor with every possible number of cycles, and we give a sufficient condition for the existence of a 2-factor in L²(G) with all cycle lengths specified. We also give a characterization of the graphs G where $L^{k} (G)$ contains a 2-factor.

The 3-path-step operator on trees and unicyclic graphs

Bohdan Zelinka (2002)

Mathematica Bohemica

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E. Prisner in his book Graph Dynamics defines the $k$ -path-step operator on the class of finite graphs. The $k$ -path-step operator (for a positive integer $k$ ) is the operator $S_{k}^{'}$ which to every finite graph $G$ assigns the graph $S_{k}^{'} (G)$ which has the same vertex set as $G$ and in which two vertices are adjacent if and only if there exists a path of length $k$ in $G$ connecting them. In the paper the trees and the unicyclic graphs fixed in the operator $S_{3}^{'}$ are studied.

Cycle and path embedding on 5-ary N-cubes

Tsong-Jie Lin, Sun-Yuan Hsieh, Hui-Ling Huang (2009)

RAIRO - Theoretical Informatics and Applications - Informatique Théorique et Applications

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We study two topological properties of the 5-ary $n$ -cube $Q_{n}^{5}$ . Given two arbitrary distinct nodes $x$ and $y$ in $Q_{n}^{5}$ , we prove that there exists an $x$ - $y$ path of every length ranging from $2 n$ to $5^{n} - 1$ , where $n \geq 2$ . Based on this result, we prove that $Q_{n}^{5}$ is 5-edge-pancyclic by showing that every edge in $Q_{n}^{5}$ lies on a cycle of every length ranging from $5$ to $5^{n}$ .

Histories in path graphs

Ludovít Niepel (2007)

Discussiones Mathematicae Graph Theory

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For a given graph G and a positive integer r the r-path graph, $P_{r} (G)$ , has for vertices the set of all paths of length r in G. Two vertices are adjacent when the intersection of the corresponding paths forms a path of length r-1, and their union forms either a cycle or a path of length k+1 in G. Let $P_{r}^{k} (G)$ be the k-iteration of r-path graph operator on a connected graph G. Let H be a subgraph of $P_{r}^{k} (G)$ . The k-history $P_{r}^{- k} (H)$ is a subgraph of G that is induced by all edges that take part in the recursive...