Related papers: Erratum for "On oriented cliques with respect to p…
We give a new unified proof that any simple graph on $n$ vertices with maximum degree at most $\Delta$ has no more than $a\binom{\Delta+1}{t}+\binom{b}{t}$ cliques of size $t \ (t \ge 3)$, where $n = a(\Delta+1)+b \ (0 \le b \le \Delta)$.
Write f(n) for the minimum number of 4-cliques in graphs of order n with triangle-free complement. Finding f(n) is a particular case of a problem raised by Erdos in 1962. We give an upper bound of f(n) and a matching lower bound when the…
By applying simplification operations to categories of multigraphs, several natural graph operations are shown to demonstrate categorical issues. The replacement of an undirected edge with a directed cycle for digraphs admits both a left…
This paper investigates some relationship between the algebraic connectivity and the clique number of graphs. We characterize all extremal graphs which have the maximum and minimum the algebraic connectivity among all graphs of order $n$…
Indeterminate strings have received considerable attention in the recent past; see for example Christodoulakis et al 2015 and Helling et al 2017. This attention is due to their applicability in bioinformatics, and to the natural…
In a directed graph, a kernel is a subset of vertices that is both stable and absorbing. Not all digraphs have a kernel, but a theorem due to Boros and Gurvich guarantees the existence of a kernel in every clique-acyclic orientation of a…
We estimate the maximum possible number of cliques of size $r$ in an $n$-vertex graph free of a fixed complete $r$-partite graph $K_{s_1, s_2, \ldots, s_r}$. By viewing every $r$-clique as a hyperedge, the upper bound on the Tur\'an number…
In the note an error in Low and Lapsley's article ("Optimization Flow Control, I: Basic Algorithm and Convergence", IEEE/ACM Transactions on Networking, 7(6), pp. 861-874, 1999) is pointed out. Because of this error the proof of the Theorem…
The celebrated theorem of Robertson and Seymour states that in the family of minor-closed graph classes, there is a unique minimal class of graphs of unbounded tree-width, namely, the class of planar graphs. In the case of tree-width, the…
In relation to oriented coloring and chromatic number, the parameter oriented relative clique number of an oriented graph $\overrightarrow{G}$, denoted by $\omega_{ro}(\overrightarrow{G})$, is the main focus of this work. We solve an open…
We amend the statement of point~(i) in Theorem~1.3 in arxiv:0901.1022 and supply the additional arguments and minor changes for the results that depend on it. We also seize the occasion and generalize to non-finitely generated lattices.
There is an error in the proof (but not the truth) of Theorem 3.2 in the author's 1985 paper "The Double-Wedge Algebra for Quantum Fields on Schwarzschild and Minkowski Spacetimes" in "Communications in Mathematical Physics". The author…
We correct a mistake in the paper with the mentioned title and reference: Calc. Var. Partial Differential Equations 58 (2019), no. 1, 58:21. It corresponds with the preprint arXiv:1712.04727.
We study the planted clique problem in which a clique of size k is planted in an Erdos-Renyi graph G(n,1/2) and one is interested in recovering this planted clique. It is widely believed that it exhibits a statistical-computational gap when…
When searching for characteristic subpatterns in potentially noisy graph data, it appears self-evident that having multiple observations would be better than having just one. However, it turns out that the inconsistencies introduced when…
Graphs whose maximum clique size exceeds half of the total number of vertices satisfy a classical property: the family of their maximum sized cliques can be pierced by a single vertex. This result dates back to a 1965 theorem by Hajnal.…
A simple graph on $n$ vertices may contain a lot of maximum cliques. But how many can it potentially contain? We will define prime and composite graphs, and we will show that if $n \ge 15$, then the grpahs with the maximum number of maximum…
The second author's $\omega$, $\Delta$, $\chi$ conjecture proposes that every graph satisties $\chi \leq \lceil \frac 12 (\Delta+1+\omega)\rceil$. In this paper we prove that the conjecture holds for all claw-free graphs. Our approach uses…
For a graph $G$ and a constant $\alpha>0$, we denote by $C_{\alpha}(G)$ the minimum order of a subgraph $H\subseteq G$ with $d(H)\ge \alpha d(G)$. Liu and Montgomery conjectured that every graph $G$ contains $K_{\Omega(t)}$ as a subdivision…
We prove that if $G$ is a sparse graph --- it belongs to a fixed class of bounded expansion $\mathcal{C}$ --- and $d\in \mathbb{N}$ is fixed, then the $d$th power of $G$ can be partitioned into cliques so that contracting each of these…