The on-line choice number of a graph is a variation of the choice number defined through a two person game. It is at least as large as the choice number for all graphs and is strictly larger for some graphs. In particular, there are graphs $G$ with $|V(G)| = 2 χ(G)+1$ whose on-line choice numbers are larger than their chromatic numbers, in contrast to a recently confirmed conjecture of Ohba that every graph $G$ with $|V(G)| \le 2 χ(G)+1$ has its choice number equal its chromatic number. Nevertheless, an on-line version of Ohba conjecture was proposed in [P. Huang, T. Wong and X. Zhu, Application of polynomial method to on-line colouring of graphs, European J. Combin., 2011]: Every graph $G$ with $|V(G)| \le 2 χ(G)$ has its on-line choice number equal its chromatic number. This paper confirms the on-line version of Ohba conjecture for graphs $G$ with independence number at most 3. We also study list colouring of complete multipartite graphs $K_{3\star k}$ with all parts of size 3. We prove that the on-line choice number of $K_{3 \star k}$ is at most $3/2k$, and present an alternate proof of Kierstead's result that its choice number is $\lceil (4k-1)/3 \rceil$. For general graphs $G$, we prove that if $|V(G)| \le χ(G)+\sqrt{χ(G)}$ then its on-line choice number equals chromatic number.
The \emph{choice number} of a graph $G$, denoted $\ch(G)$, is the minimum integer $k$ such that for any assignment of lists of size $k$ to the vertices of $G$, there is a proper colouring of $G$ such that every vertex is mapped to a colour in its list. For general graphs, the choice number is not bounded above by a function of the chromatic number. In this thesis, we prove a conjecture of Ohba which asserts that $\ch(G)=χ(G)$ whenever $|V(G)|\leq 2χ(G)+1$. We also prove a strengthening of Ohba's Conjecture which is best possible for graphs on at most $3χ(G)$ vertices, and pose several conjectu…
Let $\text{ch}(G)$ denote the choice number of a graph $G$ (also called "list chromatic number" or "choosability" of $G$). Noel, Reed, and Wu proved the conjecture of Ohba that $\text{ch}(G)=χ(G)$ when $|V(G)|\le 2χ(G)+1$. We extend this to a general upper bound: $\text{ch}(G)\le \max\{χ(G),\lceil({|V(G)|+χ(G)-1})/{3}\rceil\}$. Our result is sharp for $|V(G)|\le 3χ(G)$ using Ohba's examples, and it improves the best-known upper bound for $\text{ch}(K_{4,\dots,4})$.
A graph $G$ is called \emph{chromatic-choosable} if its choice number is equal to its chromatic number, namely $Ch(G)=χ(G)$. Ohba has conjectured that every graph $G$ satisfying $|V(G)|\leq 2χ(G)+1$ is chromatic-choosable. Since each $k$-chromatic graph is a subgraph of a complete $k$-partite graph, we see that Ohba's conjecture is true if and only if it is true for every complete multipartite graph. However, the only complete multipartite graphs for which Ohba's conjecture has been verified are: $K_{3*2,2*(k-3),1}$, $K_{3,2*(k-1)}$, $K_{s+3,2*(k-s-1),1*s}$, $K_{4,3,2*(k-4),1*2}$, and $K_{5,3,…
It was conjectured by Ohba and confirmed recently by Noel et al. that, for any graph $G$, if $|V(G)|\le 2χ(G)+1$ then $χ_l(G)=χ(G)$. This indicates that the graphs with high chromatic number are chromatic-choosable. We show that this is also the case for uniform hypergraphs and further propose a generalized version of Ohba's conjecture: for any $r$-uniform hypergraph $H$ with $r\geq 2$, if $|V(H)|\le rχ(H)+r-1$ then $χ_l(H)=χ(H)$. We show that the condition of the proposed conjecture is sharp by giving two classes of $r$-uniform hypergraphs $H$ with $|V(H)|= rχ(H)+r$ and $χ_l(H)>χ(H)$. To supp…
Ohba has conjectured \cite{ohb} that if the graph $G$ has $2χ(G)+1$ or fewer vertices then the list chromatic number and chromatic number of $G$ are equal. In this paper we prove that this conjecture is asymptotically correct. More precisely we obtain that for any $0<ε<1$, there exist an $n_0=n_0(ε)$ such that the list chromatic number of $G$ equals its chromatic number, provided $$n_0 \leq |V(G) | \le (2-ε)χ(G).$$
The dichromatic number $\vecχ(D)$ of a digraph $D$ is the least number $k$ such that the vertex set of $D$ can be partitioned into $k$ parts each of which induces an acyclic subdigraph. Introduced by Neumann-Lara in 1982, this digraph invariant shares many properties with the usual chromatic number of graphs and can be seen as the natural analog of the graph chromatic number. In this paper, we study the list dichromatic number of digraphs, giving evidence that this notion generalizes the list chromatic number of graphs. We first prove that the list dichromatic number and the dichromatic number…
It was conjectured by Ohba, and proved by Noel, Reed and Wu that $k$-chromatic graphs $G$ with $|V(G)| \le 2k+1$ are chromatic-choosable. This upper bound on $|V(G)|$ is tight: if $k$ is even, then $K_{3 \star (k/2+1), 1 \star (k/2-1)}$ and $K_{4, 2 \star (k-1)}$ are $k$-chromatic graphs with $2 k+2$ vertices that are not chromatic-choosable. It was proved in [arXiv:2201.02060] that these are the only non-$k$-choosable complete $k$-partite graphs with $2k+2$ vertices. For $G =K_{3 \star (k/2+1), 1 \star (k/2-1)}$ or $K_{4, 2 \star (k-1)}$, a bad list assignment of $G$ is a $k$-list assignment …
Let $ϕ(k)$ be the minimum number of vertices in a non-$k$-choosable $k$-chromatic graph. The Ohba conjecture, confirmed by Noel, Reed and Wu, asserts that $ϕ(k) \ge 2k+2$. This bound is tight if $k$ is even. If $k$ is odd, then it is known that $ϕ(k) \le 2k+3$ and it is conjectured by Noel that $ϕ(k) = 2k+3$. For a multi-set $λ=\{k_1,k_2, \ldots, k_q\}$ of positive integers, let $k_λ = \sum_{i=1}^q k_i$. A $λ$-list assignment of $G$ is a $k_λ$-list assignment $L$ for which the colour set $\cup_{v \in V(G)}L(v)$ can be partitioned into the disjoint union $C_1 \cup C_2 \cup \ldots \cup C_q$ of $…
Cones of Theta Graphs are Enumeratively Chromatic-Choosable
School of Computing, University of South Alabama（南阿拉巴马大学计算机学院）；McGill-Toolen Catholic High School（麦吉尔-图伦天主教高中）；Department of Mathematics and Statistics, University of South Alabama（南阿拉巴马大学数学与统计系）；Department of Applied Mathematics, Illinois Institute of Technology（伊利诺伊理工学院应用数学系）；Alabama School of Math and Science（阿拉巴马数学与科学学校）；Department of Biomedical Sciences, University of South Alabama（南阿拉巴马大学生物医学科学系）
Reed's $ω$, $Δ$, $χ$ conjecture proposes that every graph satisfies $χ\leq \lceil\frac 12(Δ+1+ω)\rceil$; it is known to hold for all claw-free graphs. In this paper we consider a local strengthening of this conjecture. We prove the local strengthening for line graphs, then note that previous results immediately tell us that the local strengthening holds for all quasi-line graphs. Our proofs lead to polytime algorithms for constructing colourings that achieve our bounds: $O(n^2)$ for line graphs and $O(n^3m^2)$ for quasi-line graphs. For line graphs, this is faster than the best known algorithm…
The linear arboricity of a graph $G$, denoted by $\text{la}(G)$, is the minimum number of edge-disjoint linear forests (i.e. forests in which every connected component is a path) in $G$ whose union covers all the edges of $G$. A famous conjecture due to Akiyama, Exoo, and Harary from 1980 asserts that $\text{la}(G)\leq \lceil (Δ(G)+1)/2 \rceil$, where $Δ(G)$ denotes the maximum degree of $G$. This conjectured upper bound would be best possible, as is easily seen by taking $G$ to be a regular graph. In this paper, we show that for every graph $G$, $\text{la}(G)\leq \fracΔ{2}+O(Δ^{2/3-α})$ for s…
A well-known combinatorial theorem says that a set of n non-collinear points in the plane determines at least n distinct lines. Chen and Chvátal conjectured that this theorem extends to metric spaces, with an appropriated definition of line. In this work we prove a slightly stronger version of Chen and Chvátal conjecture for a family of graphs containing chordal graphs and distance-hereditary graphs.
The big-line-big-clique conjecture states that for all $k,\ell\geq2$ there is an integer $n$ such that every finite set of at least $n$ points in the plane contains $\ell$ collinear points or $k$ pairwise visible points. We show that this conjecture is false for infinite point sets, by constructing a countably infinite point set with no 4 collinear points and no 3 pairwise visible points.
In 2008 Karasev conjectured that for every set of $r$ blue lines, $r$ green lines, and $r$ red lines in the plane, there exists a partition of them into $r$ colorful triples whose induced triangles intersect. We disprove this conjecture for every $r$ and extend the counterexamples to higher dimensions.
In 1968, Erdös and Lovász conjectured that for every graph $G$ and all integers $s,t\geq 2$ such that $s+t-1=χ(G) > ω(G)$, there exists a partition $(S,T)$ of the vertex set of $G$ such that $χ(G|S)\geq s$ and $χ(G|T)\geq t$. For general graphs, the only settled cases of the conjecture are when $s$ and $t$ are small. Recently, the conjecture was proved for a few special classes of graphs: graphs with stability number 2 \cite{quasi-line}, line graphs \cite{line} and quasi-line graphs \cite{quasi-line}. In this paper, we consider the conjecture for claw-free graphs and present some progress on i…
Ryser conjectured that $τ\le(r-1)ν$ for $r$-partite hypergraphs, where $τ$ is the covering number and $ν$ is the matching number. We prove this conjecture for $r\le9$ in the special case of linear intersecting hypergraphs, in other words where every pair of lines meets in exactly one vertex. Aharoni formulated a stronger version of Ryser's conjecture which specified that each $r$-partite hypergraph should have a cover of size $(r-1)ν$ of a particular form. We provide a counterexample to Aharoni's conjecture with $r=13$ and $ν=1$. We also report a number of computational results. For $r=7$, we …
Stephan (Prove or Disprove 100 Conjectures from the OES, arXiv:math/0409509v4 [math.CO])enumerates a number of conjectures regarding integer sequences contained in Sloane's On-line Encyclopedia of Integer Sequences (N. J. A. Sloane, editor, The On-Line Encyclopedia of Integer Sequences, published electronically at https://oeis.org). In this paper, we prove two of these conjectures.
We prove that for a line perfect multigraph the chromatic index is equal to the list chromatic index. This is a generalization of Galvin's result on bipartite multigraphs. Soon after the first version was submitted to arxiv, I found out that the same result has already been achieved in an older paper by Peterson and Woodall, by a similar but not entirely the same method.
We present two extensions of the linear bound, due to Marcus and Tardos, on the number of 1's in an n by n 0-1 matrix avoiding a fixed permutation matrix. We first extend the linear bound to hypergraphs with ordered vertex sets and, using previous results of Klazar, we prove an exponential bound on the number of hypergraphs on n vertices which avoid a fixed permutation. This, in turn, solves various conjectures of Klazar as well as a conjecture of Branden and Mansour.We then extend the original Furedi-Hajnal problem from ordinary matrices to d-dimensional matrices and show that the number of 1…
In this paper, we prove an enhanced version of the Erdős-Lovász Tihany Conjecture for line graphs of multigraphs. That is, for every graph $G$ whose chromatic number $χ(G)$ is more than its clique number $ω(G)$ and for nonnegative integer $\ell$, any two integers $s,t \geq 3.5\ell+2$ with $s+t = χ(G)+1$, there is a partition $(S,T)$ of the vertex set $V(G)$ such that $χ(G[S])\geq s$ and $χ(G[T])\geq t+\ell$. In particular, when $\ell=1$, we can obtain the same result just for any $s,t\geq4$. The Erdős-Lovász Tihany conjecture is a special case when $\ell=0$.
A linear forest is a forest in which every connected component is a path. The linear arboricity of a graph $G$ is the minimum number of linear forests of $G$ covering all edges. In 1980, Akiyama, Exoo and Harary proposed a conjecture, known as the Linear Arboricity Conjecture (LAC), stating that every $d$-regular graph $G$ has linear arboricity $\lceil \frac{d+1}{2} \rceil$. In 1988, Alon proved that the LAC holds asymptotically. In 1999, the list version of the LAC was raised by An and Wu, which is called the List Linear Arboricity Conjecture. In this article, we prove that the List Linear Ar…
We prove two conjectures in spectral extremal graph theory involving the linear combinations of graph eigenvalues. Let $λ_1(G)$ be the largest eigenvalue of the adjacency matrix of a graph $G$, and $\bar{G}$ be the complement of $G$. A nice conjecture states that the graph on $n$ vertices maximizing $λ_1(G) + λ_1(\bar{G})$ is the join of a clique and an independent set, with $\lfloor n/3\rfloor$ and $\lceil 2n/3\rceil$ (also $\lceil n/3\rceil$ and $\lfloor 2n/3\rfloor$ if $n \equiv 2 \pmod{3}$) vertices, respectively. We resolve this conjecture for sufficiently large $n$ using analytic methods…
The linear arboricity la(G) of a graph G is the minimum number of linear forests that partition the edges of G. In 1984, Akiyama et al. stated the Linear Arboricity Conjecture (LAC), that the linear arboricity of any simple graph of maximum degree $Δ$ is either $\lceil \tfracΔ{2} \rceil$ or $\lceil \tfrac{Δ+1}{2} \rceil$. In [J. L. Wu. On the linear arboricity of planar graphs. J. Graph Theory, 31:129-134, 1999] and [J. L. Wu and Y. W. Wu. The linear arboricity of planar graphs of maximum degree seven is four. J. Graph Theory, 58(3):210-220, 2008.] it was proven that LAC holds for all planar g…
In 2011, the second author conjectured that every line graph $G$ satisfies $χ(G)\le \max\{ω(G),\frac{5Δ(G)+8}{6}\}$. This conjecture is best possible, as shown by replacing each edge in a 5-cycle by $k$ parallel edges, and taking the line graph. In this paper we prove the conjecture. We also develop more general techniques and results that will likely be of independent interest, due to their use in attacking the Goldberg--Seymour conjecture.
In this work we present a version of the so called Chen and Chvátal's conjecture for directed graphs. A line of a directed graph D is defined by an ordered pair (u, v), with u and v two distinct vertices of D, as the set of all vertices w such that u, v, w belong to a shortest directed path in D containing a shortest directed path from u to v. A line is empty if there is no directed path from u to v. Another option is that a line is the set of all vertices. The version of the Chen and Chvátal's conjecture we study states that if none of previous options hold, then the number of distinct lines …
An $r$-uniform hypergraph ($r$-graph for short) is called linear if every pair of vertices belong to at most one edge. A linear $r$-graph is complete if every pair of vertices are in exactly one edge. The famous Brown-Erdős-Sós conjecture states that for every fixed $k$ and $r$, every linear $r$-graph with $Ω(n^2)$ edges contains $k$ edges spanned by at most $(r-2)k+3$ vertices. As an intermediate step towards this conjecture, Conlon and Nenadov recently suggested to prove its natural Ramsey relaxation. Namely, that for every fixed $k$, $r$ and $c$, in every $c$-colouring of a complete linear …
In this paper we consider a natural extremal graph theoretic problem of topological sort, concerning the minimization of the (topological) connectedness of the independence complex of graphs in terms of its dimension. We observe that the lower bound $\frac{\dim(\mathcal{I}(G))}{2} - 2$ on the connectedness of the independence complex $\mathcal{I}(G)$ of line graphs of bipartite graphs $G$ is tight. In our main theorem we characterize the extremal examples. Our proof of this characterization is based on topological machinery. Our motivation for studying this problem comes from a classical conje…
A subgraph of an edge-coloured complete graph is called rainbow if all its edges have different colours. The study of rainbow decompositions has a long history, going back to the work of Euler on Latin squares. In this paper we discuss three problems about decomposing complete graphs into rainbow trees: the Brualdi-Hollingsworth Conjecture, Constantine's Conjecture, and the Kaneko-Kano-Suzuki Conjecture. We show that in every proper edge-colouring of $K_n$ there are $10^{-6}n$ edge-disjoint spanning isomorphic rainbow trees. This simultaneously improves the best known bounds on all these conje…
Suppose that we have a set of numbers x_1, ..., x_n which have nonnegative sum. How many subsets of k numbers from {x_1, ..., x_n} must have nonnegative sum? Manickam, Miklos, and Singhi conjectured that for n at least 4k the answer is (n-1 \choose k-1). This conjecture is known to hold when n is large compared to k. The best known bounds are due to Alon, Huang, and Sudakov who proved the conjecture when n > 33k^2. In this paper we improve this bound by showing that there is a constant C such that the conjecture holds when n > Ck.
We introduce a new combinatorial abstraction for the graphs of polyhedra. The new abstraction is a flexible framework defined by combinatorial properties, with each collection of properties taken providing a variant for studying the diameters of polyhedral graphs. One particular variant has a diameter which satisfies the best known upper bound on the diameters of polyhedra. Another variant has superlinear asymptotic diameter, and together with some combinatorial operations, gives a concrete approach for disproving the Linear Hirsch Conjecture.
A generalized Terao Conjecture for line arrangements
Université Côte d’Azur, CNRS, LJAD（法国蔚蓝海岸大学）；Simion Stoilow Institute of Mathematics（罗马尼亚科学院西奥多·斯托伊洛数学研究所）；Department of Mathematics, UKEN Krakow（克拉科夫雅盖隆大学数学系）
Let A(n) be a $k\times s$ matrix and $m(n)$ be a $k$ dimensional vector, where all entries of A(n) and $m(n)$ are integer-valued polynomials in $n$. Suppose that $$t(m(n)|A(n))=#\{x\in\mathbb{Z}_{+}^{s}\mid A(n)x=m(n)\}$$ is finite for each $n\in \mathbb{N}$, where $Z_+$ is the set of nonnegative integers. This paper conjectures that $t(m(n)|A(n))$ is an integer-valued quasi-polynomial in $n$ for $n$ sufficiently large and verifies the conjecture in several cases.
Let $L_{a,b}$ be a line in the Euclidean plane with slope $a$ and intercept $b$. The dimension spectrum $\spec(L_{a,b})$ is the set of all effective dimensions of individual points on $L_{a,b}$. The dimension spectrum conjecture states that, for every line $L_{a,b}$, the spectrum of $L_{a,b}$ contains a unit interval. In this paper we prove that the dimension spectrum conjecture is true. Let $(a,b)$ be a slope-intercept pair, and let $d = \min\{\dim(a,b), 1\}$. For every $s \in (0, 1)$, we construct a point $x$ such that $\dim(x, ax + b) = d + s$. Thus, we show that $\spec(L_{a,b})$ contains t…
Karasev conjectured that for any set of $3k$ lines in general position in the plane, which is partitioned into $3$ color classes of equal size $k$, the set can be partitioned into $k$ colorful 3-subsets such that all the triangles formed by the subsets have a point in common. Although the general conjecture is false, we show that Karasev's conjecture is true for lines in convex position. We also discuss possible generalizations of this result.
Let $\mathcal{L}$ and $\mathcal{L}_0$ be the binary codes generated by the column $\mathbb{F}_2$-null space of the incidence matrix of external points versus passant lines and internal points versus secant lines with respect to a conic in $PG(2, q)$, respectively. We confirm the conjectures on the dimensions of $\mathcal{L}$ and $\mathcal{L}_0$ using methods from both finite geometry and modular representation theory.
