Journal of Graph Theory最新文献

A graph is edge-transitive if its automorphism group acts transitively on the edge set. This paper presents a complete classification for connected edge-transitive cubic graphs of order $��\; ��2�\; �n��$, where $��\; ��n��$ is even and square-free. In particular, it is shown that such a graph is either symmetric or isomorphic to one of the following graphs: a semisymmetric graph of order 420, a semisymmetric graph of order 29,260, and five families of semisymmetric graphs constructed from the simple group $��\; ��{\text{PSL}}_2�\; ��\left(�\; �p�\; �\right)���$.

In 1979, Hakimi and Schmeichel considered the problem of maximizing the number of cycles of a given length in an $��n�$-vertex planar graph. They precisely determined the maximum number of triangles and four-cycles and presented a conjecture for the maximum number of pentagons. In this work, we confirm their conjecture. Even more, we characterize the $��n�$-vertex, planar graphs with the maximum number of pentagons.

For a graph $��G�$, we define a small automorphism as one that maps some vertex into its neighbour. We investigate the edge colourings of $��G�$ that break every small automorphism of $��G�$. We show that such a colouring can be chosen from any set of lists of length 3. In addition, we show that any set of lists of length 2 on both edges and vertices of $��G�$ yields a total colouring which breaks all the small automorphisms of $��G�$. These results are sharp, and they match the known bounds for the nonlist variant.

A 3-connected graph $�\; ��G�$ is a brick if $�\; ��G�\; �-�\; �S�$ has a perfect matching, for each pair $�\; ��S�$ of vertices of $�\; ��G�$. A brick $�\; ��G�$ is minimal if $�\; ��G�\; �-�\; �e�$ ceases to be a brick for every edge $�\; ��e�\; �\in �\; �E�\; ��\left(�\; �G�\; �\right)��$. Norine and Thomas proved that each minimal brick contains at least three vertices of degree three and made a stronger conjecture: there exists $�\; ��\alpha �\; �>�\; �0�$ such that every minimal brick $�\; ��G�$ has at least $�\; ��\alpha �\; �\mid �\; �V�\; ��\left(�\; �G�\; �\right)��\; �\mid �$ cubic vertices. In this paper, we prove this conjecture holds for all minimal bricks of an average degree no less than 23/5. As its corollary, we show that each minimal brick on $�\; ��n�$ vertices contains more than $�\; ��n�\; �/�\; �5�$ vertices of degree at most four.

This corrigendum corrects an error found in the proof of correctness of the algorithm by [Ducoffe, JGT, 2022, 99(4), pp. 594–614], Theorem 6. An erroneous result from Deogun and Kratsch was used in the original proof. There are no changes in the algorithm itself.

Let $�\; ��W�$ be any wheel graph and $�\; ��G�$ the class of all countable graphs not containing $�\; ��W�$ as a minor. We show that there exists a graph in $�\; ��G�$ which contains every graph in $�\; ��G�$ as an induced subgraph.

A graph is said to be uniquely hamiltonian if it has a unique hamiltonian cycle. For a natural extension of this concept to infinite graphs, we find all uniquely hamiltonian vertex-transitive graphs with finitely many ends, and also discuss some examples with infinitely many ends. In particular, we show each nonabelian free group $�\; ��F_n�$ has a Cayley graph of degree $�\; ��2�\; �n�\; �+�\; �2�$ that has a unique hamiltonian circle. (A weaker statement had been conjectured by Georgakopoulos.) Furthermore, we prove that these Cayley graphs of $�\; ��F_n�$ are outerplanar.