Journal of Graph Theory最新文献

How many copies of a fixed odd cycle, $��C_{�2�\; �m�\; �+�\; �1�}�$, can a planar graph contain? We answer this question asymptotically for $��m�\; �\in ��\left\{��2�\; �,�\; �3�\; �,�\; �4��\; �\right\}��$ and prove a bound which is tight up to a factor of 3/2 for all other values of $��m�$. This extends the prior results of Cox and Martin and of Lv, Győri, He, Salia, Tompkins, and Zhu on the analogous question for even cycles. Our bounds result from a reduction to the following maximum likelihood question: which probability mass $��\mu �$ on the edges of some clique maximizes the probability that $��m�$ edges sampled independently from $��\mu �$ form either a cycle or a path?

A graph is called odd (respectively, even) if every vertex has odd (respectively, even) degree. Gallai proved that every graph can be partitioned into two even induced subgraphs, or into an odd and an even induced subgraph. We refer to a partition into odd subgraphs as an odd colouring of $��G�$. Scott proved that a connected graph admits an odd colouring if and only if it has an even number of vertices. We say that a graph $��G�$ is $��k�$-odd colourable if it can be partitioned into at most $��k�$ odd induced subgraphs. The odd chromatic number of $��G�$, denoted by $��{\chi }_{\text{odd}}��\left(�\; �G�\; �\right)��$, is the minimum integer $��k�$ for which $��G�$ is $��k�$-odd colourable. We initiate the systematic study of odd colouring and odd chromatic number of graph classes. We fir

In 2017, Brinkmann, Goetschalckx and Schein introduced a very general way of describing operations on embedded graphs that preserve all orientation-preserving symmetries of the graph. This description includes all well-known operations such as Dual, Truncation and Ambo. As these operations are applied locally, they are called local orientation-preserving symmetry-preserving operations (lopsp-operations). In this text, we will use the general description of these operations to determine their effect on 3-connectivity. Recently it was proved that all lopsp-operations preserve 3-connectivity of graphs that have face-width at least three. We present a simple condition that characterises exactly which lopsp-operations preserve 3-connectivity for all embedded graphs, even for those with face-width less than three.

The so-called weak-2-linkage problem asks for a given digraph $��D�=��\left(��V�,�A��\right)��$ and distinct vertices $��s_1�,�s_2�,�t_1�,�t_2�$ of $��D�$ whether $��D�$ has arc-disjoint paths $��P_1�,�P_2�$ so that $��P_i�$ is an $���(��s_{}$

For each surface besides the sphere, projective plane, and Klein bottle, we construct a face-simple minimal quadrangulation, that is, a simple quadrangulation on the fewest number of vertices possible, whose dual is also a simple graph. Our result answers a question of Liu, Ellingham, and Ye while providing a simpler proof of their main result. The inductive construction is based on an earlier idea for finding near-quadrangular embeddings of the complete graphs using the diamond sum operation.

Andreae proved that the cop number of connected -minor-free graphs is bounded for every graph . In particular, the cop number is at most if

Let be a simple graph and denote the length of the longest cycle containing an edge if there exists a cycle containing , and 2 otherwise. We prove that the summation of 2 divided by $��\; ��c_G$

We study SIR-type epidemics (susceptible-infected-resistant) on graphs in two scenarios: (i) when the initial infections start from a well-connected central region and (ii) when initial infections are distributed uniformly. Previously, Ódor et al. demonstrated on a few random graph models that the expectation of the total number of infections undergoes a switchover phenomenon; the central region is more dangerous for small infection rates, while for large rates, the uniform seeding is expected to infect more nodes. We rigorously prove this claim under mild, deterministic assumptions on the underlying graph. If we further assume that the central region has a large enough expansion, the second moment of the degree distribution is bounded and the number of initial infections is comparable to the number of vertices, the difference between the two scenarios is shown to be macroscopic.