Graphs and Combinatorics最新文献

Given a simple graph G, we ask when V(G) may be partitioned into two sets such that every vertex has an equal number of neighbors from each set. We establish a number of results for common families of graphs and completely classify 4-regular circulants which posses this property.

Abstract

A set of edges (Xsubseteq E(G)) of a graph G is an edge general position set if no three edges from X lie on a common shortest path. The edge general position number ({textrm{gp}}_{textrm{e}}(G)) of G is the cardinality of a largest edge general position set in G. Graphs G with ({textrm{gp}}_{{textrm{e}}}(G) = |E(G)| - 1) and with ({textrm{gp}}_{{textrm{e}}}(G) = 3) are respectively characterized. Sharp upper and lower bounds on ({textrm{gp}}_{{textrm{e}}}(G)) are proved for block graphs G and exact values are determined for several specific block graphs.

Given a hypergraph ({{mathcal {H}}}) and a graph G, we say that ({{mathcal {H}}}) is a Berge-G if there is a bijection between the hyperedges of ({{mathcal {H}}}) and the edges of G such that each hyperedge contains its image. We denote by (textrm{ex}_k(n,Berge- F)) the largest number of hyperedges in a k-uniform Berge-F-free graph. Let (textrm{ex}(n,H,F)) denote the largest number of copies of H in n-vertex F-free graphs. It is known that (textrm{ex}(n,K_k,F)le textrm{ex}_k(n,Berge- F)le textrm{ex}(n,K_k,F)+textrm{ex}(n,F)), thus if (chi (F)>r), then (textrm{ex}_k(n,Berge- F)=(1+o(1)) textrm{ex}(n,K_k,F)). We conjecture that (textrm{ex}_k(n,Berge- F)=textrm{ex}(n,K_k,F)) in this case. We prove this conjecture in several instances, including the cases (k=3) and (k=4). We prove the general bound (textrm{ex}_k(n,Berge- F)= textrm{ex}(n,K_k,F)+O(1)).

Let G be a graph. A subset (D subseteq V(G)) is a decycling set of G if (G-D) contains no cycle. A subset (D subseteq V(G)) is a cycle isolating set of G if (G-N[D]) contains no cycle. The decycling number and cycle isolation number of G, denoted by (phi (G)) and (iota _c(G)), are the minimum cardinalities of a decycling set and a cycle isolating set of G, respectively. Dross, Montassier and Pinlou (Discrete Appl Math 214:99–107, 2016) conjectured that if G is a planar graph of size m and girth at least g, then (phi (G) le frac{m}{g}). So far, this conjecture remains open. Recently, the authors proposed an analogous conjecture that if G is a connected graph of size m and girth at least g that is different from (C_g), then (iota _c(G) le frac{m+1}{g+2}), and they presented a proof for the initial case (g=3). In this paper, we further prove that for the cases of girth at least 4, 5 and 6, this conjecture is true. The extremal graphs of results above are characterized.

An internal or friendly partition of a graph is a partition of the vertex set into two nonempty sets so that every vertex has at least as many neighbours in its own class as in the other one. It has been shown that apart from finitely many counterexamples, every 3, 4 or 6-regular graph has an internal partition. In this note we focus on the 5-regular case and show that among the subgraphs of minimum degree at least 3 of 5-regular graphs, there are some which have small intersection. We also discuss the existence of internal partitions in some families of Cayley graphs, notably we determine all 5-regular Abelian Cayley graphs which do not have an internal partition.

In this paper, we give constructions of self-orthogonal codes from orbit matrices of Deza graphs, normally regular digraphs and Deza digraphs in terms of a definition given by Wang and Feng. These constructions can also be applied to adjacency matrices of the mentioned graphs. Since a lot of constructions of Deza graphs, normally regular digraphs and Deza digraphs in the sense of Wang and Feng have been known, the methods presented in this paper give us a rich source of matrices that span self-orthogonal codes.

We give a new elementary proof of El Sahili conjecture El Sahili (Discrete Math 287:151–153, 2004) stating that any n-chromatic digraph D, with (nge 4), contains a path with two blocks of order n.

The Gyárfás–Sumner conjecture says that for every tree T and every integer (tge 1), if G is a graph with no clique of size t and with sufficiently large chromatic number, then G contains an induced subgraph isomorphic to T. This remains open, but we prove that under the same hypotheses, G contains a subgraph H isomorphic to T that is “path-induced”; that is, for some distinguished vertex r, every path of H with one end r is an induced path of G.

In this paper, we study the relationship between spectral radius and maximum average degree of graphs. By using this relationship and the previous technique of Li and Ning in (J Graph Theory 103:486–492, 2023), we prove that, for any given positive number (varepsilon <frac{1}{3}), if n is a sufficiently large integer, then any graph G of order n with (rho (G)>sqrt{leftlfloor frac{n^{2}}{4}rightrfloor }) contains a cycle of length t for all integers (tin [3,(frac{1}{3}-varepsilon )n]), where (rho (G)) is the spectral radius of G. This improves the result of Li and Ning (2023).

A sunflower with r petals is a collection of r sets over a ground set X such that every element in X is in no set, every set, or exactly one set. Erdős and Rado [5] showed that a family of sets of size n contains a sunflower if there are more than (n!(r-1)^n) sets in the family. Alweiss et al. [1] and subsequently, Rao [7] and Bell et al. [2] improved this bound to ((O(r log n))^n). We study the case where the pairwise intersections of the set family are restricted. In particular, we improve the best known bound for set families when the size of the pairwise intersections of any two sets is in a set L. We also present a new bound for the special case when the set L is the nonnegative integers less than or equal to d using the techniques of Alweiss et al. [1].