Journal of Combinatorial Theory Series B最新文献

A $(4,5)$-coloring of $K_n$ is an edge-coloring of $K_n$ where every 4-clique spans at least five colors. We show that there exist $(4,5)$-colorings of $K_n$ using $\frac{5}{6}n+o\left(n\right)$ colors. This settles a disagreement between Erdős and Gyárfás reported in their 1997 paper. Our construction uses a randomized process which we analyze using the so-called differential equation method to establish dynamic concentration. In particular, our coloring process uses random triangle removal, a process first introduced by Bollobás and Erdős, and analyzed by Bohman, Frieze and Lubetzky.

We prove that for any graph G of maximum degree at most Δ, the zeros of its chromatic polynomial ${\chi }_G\left(x\right)$ (in $C$) lie inside the disc of radius 5.94Δ centered at 0. This improves on the previously best known bound of approximately 6.91Δ.

We also obtain improved bounds for graphs of high girth. We prove that for every g there is a constant $K_g$ such that for any graph G of maximum degree at most Δ and girth at least g, the zeros of its chromatic polynomial ${\chi }_G\left(x\right)$ lie inside the disc of radius $K_g\Delta$ centered at 0, where $K_g$ is the solution to a certain optimization problem. In particular, $K_g<5$ when $g\ge 5$ and $K_g<4$ when $g\ge 25$ and $K_g$ tends to approximately 3.86 as $g\to \infty$.

Key to the proof is a classical theorem of Whitney which allows us to relate the chromatic polynomial of a graph G to the generating function of so-called broken-circuit-free forests in G. We also establish a zero-free disc for the generating function of all forests in G (aka the partition function of the arboreal gas) which may be of independent interest.

A graph G is $(2,m)$-linked if, for any distinct vertices $a_1,\dots ,a_m,b_1,b_2$ in G, there exist disjoint connected subgraphs $A,B$ of G such that $a_1,\dots ,a_m\in V\left(A\right)$ and $b_1,b_2\in V\left(B\right)$. A fundamental result in structural graph theory is the characterization of $(2,2)$-linked graphs. It appears to be difficult to characterize $(2,m)$-linked graphs for $m\ge 3$. In this paper, we provide a partial characterization of $(2,m)$-linked graphs. This implies that every $(2m+2)$-connected graphs G is $(2,m)$-linked and for any distinct vertices $a_1,\dots ,a_m,b_1,b_2$ of G, there is a path P in G between $b_1$ and $b_2$ and avoiding $\{a_1,\dots ,a_m\}$ such that $G-P$ is connected, improving a previous connectivity bound of 10m.

Given a graph G, consider the family of real symmetric matrices with the property that the pattern of their nonzero off-diagonal entries corresponds to the edges of G. For the past 30 years a central problem has been to determine which spectra are realizable in this matrix class. Using combinatorial methods, we identify a family of graphs and multiplicity lists whose realizable spectra are highly restricted. In particular, we construct trees with multiplicity lists that require a unique spectrum, up to shifting and scaling. This represents the most extreme possible failure of spectral arbitrariness for a multiplicity list, and greatly extends all previously known instances of this phenomenon, in which only single linear constraints on the eigenvalues were observed.

Let $H_k^r$ denote an r-uniform hypergraph with k edges and $r+1$ vertices, where $k\le r+1$ (it is easy to see that such a hypergraph is unique up to isomorphism). The known general bounds on its Turán density are $\pi \left(H_k^r\right)\le \frac{k-2}{r}$ for all $k\ge 3$, and $\pi \left(H_3^r\right)\ge 2^{1-r}$ for $k=3$. We prove that $\pi \left(H_k^r\right)\ge (C_k-o(1\left)\right)r^{-(1+\frac{1}{k-2})}$ as $r\to \infty$. In the case $k=3$, we prove $\pi \left(H_3^r\right)\ge (1.7215-o(1\left)\right)r^{-2}$ as $r\to \infty$, and $\pi \left(H_3^r\right)\ge r^{-2}$ for all r.

Given a graph H, let $g(n,H)$ denote the smallest k for which the following holds. We can assign a k-colouring $f_v$ of the edge set of $K_n$ to each vertex v in $K_n$ with the property that for any copy T of H in $K_n$, there is some $u\in V\left(T\right)$ such that every edge in T has a different colour in $f_u$.

The study of this function was initiated by Alon and Ben-Eliezer. They characterized the family of graphs H for which $g(n,H)$ is bounded and asked whether it is true that for every other graph $g(n,H)$ is polynomial. We show that this is not the case and characterize the family of connected graphs H for which $g(n,H)$ grows polynomially. Answering another question of theirs, we also prove that for every $\epsilon >0$, there is some $r=r\left(\epsilon \right)$ such that $g(n,K_r)\ge n^{1-\epsilon }$ for all sufficiently large n.

Finally, we show that the above problem is connected to the Erdős–Gyárfás function in Ramsey Theory, and prove a family of special cases of a conjecture of Conlon, Fox, Lee and Sudakov by showing that for each fixed r the complete r-uniform hypergraph $K_n^{\left(r\right)}$ can be edge-coloured using a subpolynomial number of colours in such a way that at least r colours appear among any $r+1$ vertices.

We study the connections between the notions of combinatorial discrepancy and graph degeneracy. In particular, we prove that the maximum discrepancy over all subgraphs H of a graph G of the neighborhood set system of H is sandwiched between $\Omega (\log \mathrm{deg}(G\left)\right)$ and $O\left(\mathrm{deg}\right(G\left)\right)$, where $\mathrm{deg}\left(G\right)$ denotes the degeneracy of G. We extend this result to inequalities relating weak coloring numbers and discrepancy of graph powers and deduce a new characterization of bounded expansion classes.

Then we switch to a model theoretical point of view, introduce pointer structures, and study their relations to graph classes with bounded expansion. We deduce that a monotone class of graphs has bounded expansion if and only if all the set systems definable in this class have bounded hereditary discrepancy.

Using known bounds on the VC-density of set systems definable in nowhere dense classes we also give a characterization of nowhere dense classes in terms of discrepancy.

As consequences of our results, we obtain a corollary on the discrepancy of neighborhood set systems of edge colored graphs, a polynomial-time algorithm to compute ε-approximations of size $O(1/\epsilon )$ for set systems definable in bounded expansion classes, an application to clique coloring, and even the non-existence of a quantifier elimination scheme for nowhere dense classes.

A graph G is q-Ramsey for a q-tuple of graphs $(H_1,\dots ,H_q)$ if for every q-coloring of the edges of G there exists a monochromatic copy of $H_i$ in color i for some $i\in \left[q\right]$. Over the last few decades, researchers have investigated a number of questions related to this notion, aiming to understand the properties of graphs that are q-Ramsey for a fixed tuple. Among the tools developed while studying questions of this type are gadget graphs, called signal senders and determiners, which have proven invaluable for building Ramsey graphs with certain properties. However, until now these gadgets have been shown to exist and used mainly in the two-color setting or in the symmetric multicolor setting, and our knowledge about their existence for multicolor asymmetric tuples is extremely limited. In this paper, we construct such gadgets for any tuple of cliques. We then use these gadgets to generalize three classical theorems in this area to the asymmetric multicolor setting.

Given a tree T of order n, one can contract any edge and obtain a new tree $T^{⁎}$ of order $n-1$. In 1983, Jamison made a conjecture that the mean subtree order, i.e., the average order of all subtrees, decreases at least $\frac{1}{3}$ in contracting an edge of a tree. In 2023, Luo, Xu, Wagner and Wang proved the case when the edge to be contracted is a pendant edge. In this article, we prove that the conjecture is true in general.

A well-known theorem of Sabidussi shows that a simple G-arc-transitive graph can be represented as a coset graph for the group G. This pivotal result is the standard way to turn problems about simple arc-transitive graphs into questions about groups. In this paper, the Sabidussi representation is extended to arc-transitive, not necessarily simple graphs which satisfy a local-finiteness condition: namely graphs with finite valency and finite edge-multiplicity. The construction yields a G-arc-transitive coset graph $\mathrm{Cos}(G,H,J)$, where $H,J$ are stabilisers in G of a vertex and incident edge, respectively. A first major application is presented concerning arc-transitive maps on surfaces: given a group $G=〈a,z〉$ with $\left|z\right|=2$ and $\left|a\right|$ finite, the coset graph $\mathrm{Cos}(G,〈a〉,〈z〉)$ is shown, under suitable finiteness assumptions, to have exactly two different arc-transitive embeddings as a G-arc-transitive map $(V,E,F)$ (with $V,E,F$ the sets of vertices, edges and faces, respectively), namely, a G-rotary map if $\left|az\right|$ is finite, and a G-bi-rotary map if $\left|z{z}^a\right|$ is finite. The G-rotary map can be represented as a coset geometry for G, extending the notion of a coset graph. However the G-bi-rotary map does not have such a representation, and the face boundary cycles must be specified in addition to incidences between faces and edges. In addition a coset geometry construction is given of a flag-regular map $(V,E,F)$ for non necessarily simple graphs. For all of these constructions it is proved that the face boundary cycles are simple cycles precisely when the given group acts faithfully on $V\cup F$. Illustrative examples are given for graphs related to the n-dimensional hypercubes and the Petersen graph.