An independent set of size in a finite undirected graph is a set of vertices of the graph, no two of which are connected by an edge. Let be the number of independent sets of size in the graph and let
An independent set of size in a finite undirected graph is a set of vertices of the graph, no two of which are connected by an edge. Let be the number of independent sets of size in the graph and let
Carmesin and Gollin proved that every finite graph has a canonical tree-decomposition of adhesion less than that efficiently distinguishes every two distinct -profiles, and which has the further property that every separable -block is equal to the unique part of in which it is contained. We give a shorter proof of this result by showing that such a tree-decomposition can in fact be obtained from any canonical tight tree-decomposition of adhesion less than . For this, we decompose the parts of such a tree-decomposition by further tree-decompositions. As an application, we also obtain a generalization of Carmesin and Gollin's result to locally finite graphs.
One of the most fundamental results in graph theory is Mantel's theorem which determines the maximum number of edges in a triangle-free graph of order . Recently, a colorful variant of this problem has been solved. In this variant we consider graphs on a common vertex set, think of each graph as edges in a distinct color, and want to determine the smallest number of edges in each color which guarantees the existence of a rainbow triangle. Here, we solve the analogous problem for directed graphs without rainbow triangles, either directed or transitive, for any number of colors. The constructions and proofs essentially differ for and and the type of the forbidden triangle. Additionally, we also solve the analogous problem in the setting of oriented graphs.
Fiol, Garriga, and Yebra introduced the notion of pseudo-distance-regular vertices, which they used to come up with a new characterization of distance-regular graphs. Building on that work, Fiol and Garriga developed the spectral excess theorem for distance-regular graphs. We extend both these characterizations to distance-biregular graphs and show how these characterizations can be used to study bipartite graphs with distance-regular halved graphs and graphs with the spectrum of a distance-biregular graph.
We restate Thomassen's theorem of 3-extendability (Thomassen, Journal of Combinatorial Theory Series B, 97, 571–583), an extension of the famous planar 5-choosability theorem, in terms of graph polynomials. This yields an Alon–Tarsi equivalent of 3-extendability.
A subgraph of a multigraph is overfull if Analogous to the Overfull Conjecture proposed by Chetwynd and Hilton in 1986, Stiebitz et al. formed the multigraph version of the conjecture as follows: Let be a multigraph with maximum multiplicity and maximum degree
We determine the maximum number of copies of in a -free -vertex graph for all integers and sufficiently large . Moreover, for and any integer , we obtain the maximum number of cycles of length in an -vertex -free bipartite graph.
In this paper, we study the maximum degree in nonempty-induced subgraphs of the Kneser graph . One of the main results asserts that, for and , whenever a nonempty subgraph has
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