Journal of Combinatorial Theory Series B最新文献

We resolve a conjecture of Cox and Martin by determining asymptotically for every $k\ge 2$ the maximum number of copies of $C_{2k}$ in an n-vertex planar graph.

The Erdős-Sós conjecture states that the maximum number of edges in an n-vertex graph without a given k-vertex tree is at most $\frac{n(k-2)}{2}$. Despite significant interest, the conjecture remains unsolved. Recently, Caro, Patkós, and Tuza considered this problem for host graphs that are connected. Settling a problem posed by them, for a k-vertex tree T, we construct n-vertex connected graphs that are T-free with at least $(1/4-o_k(1\left)\right)nk$ edges, showing that the additional connectivity condition can reduce the maximum size by at most a factor of 2. Furthermore, we show that this is optimal: there is a family of k-vertex brooms T such that the maximum size of an n-vertex connected T-free graph is at most $(1/4+o_k(1\left)\right)nk$.

We show that for every graph without isolated edge, the edges can be assigned weights from $\{1,2,3\}$ so that no two neighbors receive the same sum of incident edge weights. This solves a conjecture of Karoński, Łuczak, and Thomason from 2004.

The graph removal lemma is a fundamental result in extremal graph theory which says that for every fixed graph H and $\epsilon >0$, if an n-vertex graph G contains $\epsilon n^2$ edge-disjoint copies of H then G contains $\delta n^{v\left(H\right)}$ copies of H for some $\delta =\delta (\epsilon ,H)>0$. The current proofs of the removal lemma give only very weak bounds on $\delta (\epsilon ,H)$, and it is also known that $\delta (\epsilon ,H)$ is not polynomial in ε unless H is bipartite. Recently, Fox and Wigderson initiated the study of minimum degree conditions guaranteeing that $\delta (\epsilon ,H)$ depends polynomially or linearly on ε. In this paper we answer several questions of Fox and Wigderson on this topic.

A simple graph G with maximum degree Δ is overfull if $\left|E\right(G\left)\right|>\Delta \lfloor \left|V\right(G\left)\right|/2\rfloor$. The core of G, denoted $G_{\Delta }$, is the subgraph of G induced by its vertices of degree Δ. Clearly, the chromatic index of G equals $\Delta +1$ if G is overfull. Conversely, Hilton and Zhao in 1996 conjectured that if G is a simple connected graph with $\Delta \ge 3$ and $\Delta \left(G_{\Delta }\right)\le 2$, then ${\chi }^{\prime }\left(G\right)=\Delta +1$ implies that G is overfull or $G=P^{⁎}$, where $P^{⁎}$ is obtained from the Petersen graph by deleting a vertex. Cariolaro and Cariolaro settled the base case $\Delta =3$ in 2003, and Cranston and Rabern proved the next case, $\Delta =4$, in 2019. In this paper, we give a proof of this conjecture for all $\Delta \ge 4$.

In this paper we investigate orders, longest cycles and the number of cycles of automorphisms of finite vertex-transitive graphs. In particular, we show that the order of every automorphism of a connected vertex-transitive graph with n vertices and of valence d, $d\le 4$, is at most $c_{d}n$ where $c_3=1$ and $c_4=9$. Whether such a constant $c_d$ exists for valencies larger than 4 remains an unanswered question. Further, we prove that every automorphism g of a finite connected 3-valent vertex-transitive graph Γ, $\Gamma \ncong K_{3,3}$, has a regular orbit, that is, an orbit of $〈g〉$ of length equal to the order of g. Moreover, we prove that in this case either Γ belongs to a well understood family of exceptional graphs or at least 5/12 of the vertices of Γ belong to a regular orbit of g. Finally, we give an upper bound on the number of orbits of a cyclic group of automorphisms C of a connected 3-valent vertex-transitive graph Γ in terms of the number of vertices of Γ and the length of a longest orbit of C.

A graph H is common if the number of monochromatic copies of H in a 2-edge-colouring of the complete graph $K_n$ is asymptotically minimised by the random colouring, or equivalently, $t_H\left(W\right)+t_H(1-W)\ge 2^{1-e\left(H\right)}$ holds for every graphon $W:{[0,1]}^2\to [0,1]$, where $t_H(.)$ denotes the homomorphism density of the graph H. Paths and cycles being common is one of the earliest cornerstones in extremal graph theory, due to Mulholland and Smith (1959), Goodman (1959), and Sidorenko (1989).

We prove a graph homomorphism inequality that extends the commonality of paths and cycles. Namely, $t_H\left(W\right)+t_H(1-W)\ge t_{K_2}{\left(W\right)}^{e\left(H\right)}+t_{K_2}{(1-W)}^{e\left(H\right)}$ whenever H is a path or a cycle and $W:{[0,1]}^2\to R$ is a bounded symmetric measurable function.

This answers a question of Sidorenko from 1989, who proved a slightly weaker result for even-length paths to prove the commonality of odd cycles. Furthermore, it also settles a recent conjecture of Behague, Morrison, and Noel in a strong form, who asked if the inequality holds for graphons W and odd cycles H. Our proof uses Schur convexity of complete homogeneous symmetric functions, which may be of independent interest.

The biclique partition number of a graph $G=(V,E)$, denoted $bp\left(G\right)$, is the minimum number of pairwise edge disjoint complete bipartite subgraphs of G so that each edge of G belongs to exactly one of them. It is easy to see that $bp\left(G\right)\le n-\alpha \left(G\right)$, where $\alpha \left(G\right)$ is the maximum size of an independent set of G. Erdős conjectured in the 80's that for almost every graph G equality holds; i.e., if $G=G_{n,1/2}$ then $bp\left(G\right)=n-\alpha \left(G\right)$ with high probability. Alon showed that this is false. We show that the conjecture of Erdős is true if we instead take $G=G_{n,p}$, where p is constant and less than a certain threshold value $p_0\approx 0.312$. This verifies a conjecture of Chung and Peng for these values of p. We also show that if $p_0<p<1/2$ then $bp\left(G_{n,p}\right)=n-(1+\Theta (1\left)\right)\alpha \left(G_{n,p}\right)$ with high probability.