The optimal packings of n unit discs in the plane are known for those natural numbers n, which satisfy certain number theoretic conditions. Their geometric realizations are the extremal Groemer packings (or Wegner packings). But an extremal Groemer packing of n unit discs does not exist for all natural numbers n and in this case, the number n is called exceptional. We are interested in number theoretic characterizations of the exceptional numbers. A counterexample is given to a conjecture of Wegner concerning such a characterization. We further give a characterization of the exceptional numbers, whose shape is closely related to that of Wegner's conjecture.
Thomassen formulated the following conjecture: Every $3$-connected cubic graph has a red-blue vertex coloring such that the blue subgraph has maximum degree $1$ (that is, it consists of a matching and some isolated vertices) and the red subgraph has minimum degree at least $1$ and contains no $3$-edge path. We prove the conjecture for Generalized Petersen graphs. We indicate that a coloring with the same properties might exist for any subcubic graph. We confirm this statement for all subcubic trees.
Counterexamples to Wegner's Conjecture for Rectangles
University College Dublin（都柏林大学学院）；Center for Quantum and Topological Systems, New York University Abu Dhabi（阿布扎比纽约大学量子与拓扑系统中心）；Indraprastha Institute of Information Technology Delhi（德里因德拉普拉斯塔信息技术学院）；New York University Abu Dhabi（阿布扎比纽约大学）
The Frankl conjecture (called also union-closed sets conjecture) is one of the famous unsolved conjectures in combinatorics of finite sets. In this short note, we introduce and to some extent justify some variants of the Frankl conjecture.
We study analogues of Sidorenko's conjecture and the forcing conjecture in oriented graphs, showing that natural variants of these conjectures in directed graphs are equivalent to the asymmetric, undirected analogues of the conjectures.
We prove that $\min\{χ(G), χ(H)\} - χ(G\times H)$ can be arbitrarily large, and that if Stahl's conjecture on the multichromatic number of Kneser graphs holds, then $\min\{χ(G), χ(H)\}/χ(G\times H) \leq 1/2 + ε$ for large values of $\min\{χ(G), χ(H)\}$.
In this note, we show that a "Toy Conjecture" made by (Boyle, Ishai, Pass, Wootters, 2017) is false, and propose a new one. Our attack does not falsify the full ("non-toy") conjecture in that work, and it is our hope that this note will help further the analysis of that conjecture. Independently, (Boyle, Holmgren, Ma, Weiss, 2021) have obtained similar results.
Norine's antipodal-colouring conjecture, in a form given by Feder and Subi, asserts that whenever the edges of the discrete cube are 2-coloured there must exist a path between two opposite vertices along which there is at most one colour change. The best bound to date was that there must exist such a path with at most $n/2$ colour changes. Our aim in this note is to improve this upper bound to $(\frac{3}{8}+o(1))n$.
Let $G=(V(G), E(G))$ be a multigraph with maximum degree $Δ(G)$, chromatic index $χ'(G)$ and total chromatic number $χ''(G)$. The Total Coloring conjecture proposed by Behzad and Vizing, independently, states that $χ''(G)\leq Δ(G)+μ(G) +1$ for a multigraph $G$, where $μ(G)$ is the multiplicity of $G$. Moreover, Goldberg conjectured that $χ''(G)=χ'(G)$ if $χ'(G)\geq Δ(G)+3$ and noticed the conjecture holds when $G$ is an edge-chromatic critical graph. By assuming the Goldberg-Seymour conjecture, we show that $χ''(G)=χ'(G)$ if $χ'(G)\geq \max\{ Δ(G)+2, |V(G)|+1\}$ in this note. Consequently, $χ'…
Two new sufficient conditions for generalized cycles (including Hamilton and dominating cycles as special cases) in an arbitrary k-connected graph (k=1,2,...) are derived, which prove the truth of Bondy's (1980) famous conjecture for some variants significantly improving the result expected by the given hypothesis. Similarly, two new lower bounds for the circumference (the length of a longest cycle) are established for the reverse hypothesis proposed by Jung (2001).
The Gyárfás-Sumner conjecture says that for every tree $T$ and every integer $t\ge 1$, 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 $G$.
The 1-2-3 Conjecture, posed by Karoński, Łuczak and Thomason, asked whether every connected graph $G$ different from $K_2$ can be 3-edge-weighted so that every two adjacent vertices of $G$ get distinct sums of incident weights. The 1-2 Conjecture states that if vertices also receive colors and the vertex color is added to the sum of its incident edges, then adjacent vertices can be distinguished using only $\{ 1,2\}$. In this paper we confirm 1-2 Conjecture for 3-regular graphs. Meanwhile, we show that every 3-regular graph can achieve a neighbor sum distinguishing edge coloring by using 4 col…
The road colouring theorem characterizes the class of strongly connected directed graphs with constant out-degree that admit a synchronizing road colouring. The subject of this paper is a pair of related conjectures that generalize the road colouring theorem to graphs with non-constant out-degree; we give reasons to believe that both of these conjectures are true. Our main results focus on two classes of graphs, proving both conjectures for one class of graphs and one of the conjectures for an additional class of graphs. We also present computer simulations that give some empirical evidence fo…
Let $γ(G)$ denote the domination number of graph $G$. Let $G$ and $H$ be graphs and $G\Box H$ their Cartesian product. For $h\in V(H)$ define $G_h=\{(g,h)\,|\,g\in V(G)\}$ and call this set a $G$-layer of $G\Box H$. We prove the following special case of Vizing's conjecture. Let $D$ be a dominating set of $G\Box H$. If there exist minimum dominating sets $D_1$ and $D_2$ of $G$ such that for every $h\in V(H)$, the projection of $D\cap G_h$ to $G$ is contained in $D_1$ or $D_2$, then $|D|\geq γ(G)γ(H)$.
Neumann-Lara and Škrekovski conjectured that every planar digraph is $2$-colourable. We show that this conjecture is equivalent to the more general statement that all oriented $K_5$-minor-free graphs are $2$-colourable.
A conjecture of Borwein asserts that for any positive integers $n$ and $k$, the coefficient $a_{3k}$ of $q^{3k}$ in the expansion of $\prod_{j=0}^n (1-q^{3j+1})(1-q^{3j+2})$ is nonnegative. In this paper we prove that for any $0 \leq k\leq n$, there is a constant $0<c<1$ such that $$a_{3k}+a_{3(n+1)+3k}+\cdots+a_{3n(n+1)+3k}=\frac {2\cdot 3^{n}} {n+1}(1+O(c^n)).$$ In particular, $$a_{3k}+a_{3(n+1)+3k}+\cdots+a_{3n(n+1)+3k}>0.$$
