Graphs and Combinatorics最新文献

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].

Abstract

The planar Turán number of a graph H, denoted by (ex_{_mathcal {P}}(n,H)) , is the maximum number of edges in a planar graph on n vertices without containing H as a subgraph. This notion was introduced by Dowden in 2016 and has attracted quite some attention since then; those work mainly focus on finding (ex_{_mathcal {P}}(n,H)) when H is a cycle or Theta graph or H has maximum degree at least four. In this paper, we first completely determine the exact values of (ex_{_mathcal {P}}(n,H)) when H is a cubic graph. We then prove that (ex_{_mathcal {P}}(n,2C_3)=lceil 5n/2rceil -5) for all (nge 6) , and obtain the lower bounds of (ex_{_mathcal {P}}(n,2C_k)) for all (nge 2kge 8) . Finally, we also completely determine the exact values of (ex_{_mathcal {P}}(n,K_{2,t})) for all (tge 3) and (nge t+2) .

A graph is k-vertex-critical if (chi (G)=k) but (chi (G-v)<k) for all (vin V(G)). We construct new infinite families of k-vertex-critical ((P_5,C_5))-free graphs for all (kge 6). Our construction generalises known constructions for 4-vertex-critical (P_7)-free graphs and 5-vertex-critical (P_5)-free graphs and is in contrast to the fact that there are only finitely many 5-vertex-critical ((P_5,C_5))-free graphs. In fact, our construction is even more well-structured, being ((2P_2,K_3+P_1,C_5))-free.

According to Ramsey’s Theorem, for any natural p and q there is a minimum number R(p, q) such that every graph with at least R(p, q) vertices has either a clique of size p or an independent set of size q. In the present paper, we study Ramsey numbers R(p, q) for graphs in special classes. It is known that for graphs of bounded co-chromatic number Ramsey numbers are upper-bounded by a linear function of p and q. However, the exact values of R(p, q) are known only for classes of graphs of co-chromatic number at most 2. In this paper, we determine the exact values of Ramsey numbers for classes of graphs of co-chromatic number at most 3. It is also known that for classes of graphs of unbounded splitness the value of R(p, q) is lower-bounded by ((p-1)(q-1)+1). This lower bound coincides with the upper bound for perfect graphs and for all their subclasses of unbounded splitness. We call a class Ramsey-perfect if there is a constant c such that (R(p,q)=(p-1)(q-1)+1) for all (p,qge c) in this class. In the present paper, we identify a number of Ramsey-perfect classes which are not subclasses of perfect graphs.

The k-weak-dynamic number of a graph G is the smallest number of colors we need to color the vertices of G in such a way that each vertex v of degree d(v) sees at least min({k,d(v)}) colors on its neighborhood. We use reducible configurations and list coloring of graphs to prove that all planar graphs have 3-weak-dynamic number at most 6.