A hole in a graph is a chordless cycle of length at least 4. A theta is a graph formed by three paths between the same pair of distinct vertices so that the union of any two of the paths induces a hole. A wheel is a graph formed by a hole and a node that has at least 3 neighbors in the hole. In this paper we obtain a decomposition theorem for the class of graphs that do not contain an induced s...

the main goal of this paper is to introduce and study a new class of function via the notions of $e$-$theta$-open sets and $e$-$theta$-closure operator which are defined by özkoç and aslım cite{10} called weakly $er$-open functions and $e$-$theta$-open functions. moreover, we investigate not only some of their basic properties but also their relationships with other types of already existing to...

For any h ∈ N, a graph G = (V, E) is said to be h-magic if there exists a labeling l : E(G) → Zh−{0} such that the induced vertex labeling l+ : V (G) → Zh defined by l(v) = ∑ uv∈E(G) l(uv) is a constant map. When this constant is 0 we call G a zero-sum h-magic graph. The null set of G is the set of all natural numbers h ∈ N for which G admits a zero-sum h-magic labeling. A graph G is said to be...

We show that for every rational number $r \in (1,2)$ of the form $2 - a/b$, where $a, b \mathbb{N}^+$ satisfy $\lfloor b/a \rfloor^3 \le a / (\lfloor \rfloor +1) + 1$, there exists graph $F_r$ such Tur\'an $\operatorname{ex}(n, F_r) = \Theta(n^r)$. Our result in particular generates infinitely many new exponents. As byproduct, we formulate framework is taking shape recent work on Bukh--Conlon c...

Abstract The theta body of a graph, introduced by Grötschel, Lovász, and Schrijver (in 1986), is tractable relaxation the independent-set polytope derived from Lovász number. In this paper, we recursively extend body, hence number, to hypergraphs. We obtain fundamental properties extension relate it high-dimensional Hoffman bound Filmus, Golubev, Lifshitz. discuss two applications: triangle-fre...

We generalise some well-known graph parameters to operator systems by considering their underlying quantum channels. In particular, we introduce the quantum complexity as the dimension of the smallest co-domain Hilbert space a quantum channel requires to realise a given operator system as its non-commutative confusability graph. We describe quantum complexity as a generalised minimum semidefini...

We give essentially tight bounds for, $\nu(d,k)$, the maximum number of distinct neighbourhoods on a set $X$ $k$ vertices in graph with twin-width at most~$d$. Using celebrated Marcus-Tardos theorem, two independent works [Bonnet et al., Algorithmica '22; Przybyszewski '22] have shown upper bound $\nu(d,k) \leqslant \exp(\exp(O(d)))k$, double-exponential dependence twin-width. The work [Gajarsk...

If G is a triangle-free graph, then two Gallai identities can be written as α(G)+ χ(L(G)) = |V (G)| = α(L(G))+ χ(G), where α and χ denote the stability number and the clique-partition number, and L(G) is the line graph of G. We show that, surprisingly, both equalities can be preserved for any graph G by deleting the edges of the line graph corresponding to simplicial pairs of adjacent arcs, acc...

Abstract A long-standing conjecture of Erd?s and Simonovits asserts that for every rational number $r\in (1,2)$ there exists a bipartite graph H such $\mathrm{ex}(n,H)=\Theta(n^r)$ . So far this is known to be true only rationals form $1+1/k$ $2-1/k$ , integers $k\geq 2$ In paper, we add new which the true: $2-2/(2k+1)$ This in turn also gives an affirmative answer question Pinchasi Sharir on c...