#### 28 Citations

The Line Graph of the Universal Homogeneous Triangle-free Graph

- Mathematics
- 2016

We present some results on the structure of line graphs of triangle-free graphs and a homogenization of the line graph of the universal homogeneous triangle-free graph in a finite relational language… Expand

On Asymmetric Colourings of Claw-Free Graphs

- Computer Science
- Electron. J. Comb.
- 2021

It is proved that if G is a connected, countable graph of maximum degree at most 4, without an induced claw K1,3, then D(G) = 2 whenever m (G) > 2, with three exceptional small graphs, answering the question of Babai for d = 4 in the class of claw-free graphs. Expand

Whitney's Theorem for Line Graphs of Multi-Graphs

- Mathematics
- 2021

Whitney’s Theorem states that every graph, different from K3 or K1,3, is uniquely determined by its line graph. A 1-line graph of a multigraph is the graph with as vertices the edges of the… Expand

Distinguishing numbers of finite 4-valent vertex-transitive graphs

- Mathematics, Computer Science
- Ars Math. Contemp.
- 2020

The distinguishing number of a graph $G$ is the smallest $k$ such that $G$ admits a $k$-colouring for which the only colour-preserving automorphism of $G$ is the identity. We determine the… Expand

A poset metric from the directed maximum common edge subgraph

- Computer Science, Mathematics
- ArXiv
- 2019

The proof that there is a metric based on DMCES involves the extension of the concept of line digraphs, which can be used directly to compute the metric, but is computationally feasible only for sparse graphs. Expand

Breaking graph symmetries by edge colourings

- Mathematics, Computer Science
- J. Comb. Theory, Ser. B
- 2017

It is proved that if every non-trivial automorphism of a countable graph G moves infinitely many edges, then D(G) = 2, which is the least number of colours needed in an edge colouring which is not preserved by any non-Trivial Automorphism. Expand

Line graphs of triangle-free graphs

- 2015

We present some results on the structure of line graphs of triangle-free graphs and discuss connections with homogeneity.

Uniquely Derived Generalized Line Graphs and Uniquely Represented Signed Graphs with Least Eigenvalues $$\ge $$≥$$-2$$-2

- Computer Science, Mathematics
- Graphs Comb.
- 2015

Two results are derived which are related to Cameron’s result; one is on generalized line graphs and the other is on a class of signed graphs with least eigenvalues which generalizes the following one due to A. Torga s̆ ev: If the most eigenvalue of a connected countably infinite graph is at least $-2$$-2, then it is a generalized line graph. Expand

Domination in Graphs

- Mathematics
- 2012

“Domination in graphs” is an area of graph theory that has received a lot of attention in recent years. It is reasonable to believe that “domination in graphs” has its origin in “chessboard… Expand

Eulerian and Hamiltonian Graphs

- Mathematics
- 2012

The study of Eulerian graphs was initiated in the 18th century and that of Hamiltonian graphs in the 19th century. These graphs possess rich structures; hence, their study is a very fertile field of… Expand

#### References

SHOWING 1-2 OF 2 REFERENCES

Theory of Graphs

- Mathematics
- 1962

Fundamental concepts Connectedness Path problems Trees Leaves and lobes The axiom of choice Matching theorems Directed graphs Acyclic graphs Partial order Binary relations and Galois correspondences… Expand

Congruent Graphs and the Connectivity of Graphs

- Mathematics
- 1932

We give here conditions that two graphs be congruent and some theorems on the connectivity of graphs, and we conclude with some applications to dual graphs. These last theorems might also be proved… Expand