Super edge-magic labeling of graphs: deficiency and maximality

Baskoro et al. [1] provided some constructions of new super edge-magic graphs from some old ones by attaching 1, 2, or 3 pendant vertices and edges. In this paper, we introduce (q, m)-super edge-magic total labeling and we give a construction of new super edge-magic graphs by attaching n pendant vertices and edges under some conditions, for any positive integer n. Also, we give a constraint on our construction.

New families of graphs without short cycles and large size

The problem of finding the induced subgraph with the maximum number of edges among all subsets of a fixed cardinality is known as the maximum subgraph problem (MSP). The lexicographic order has been proven optimal for Cartesian products of specific regular graphs [B. Sergei, B. Pavle and K. Nikola, New infinite family of regular edge isoperimetric graphs, Theoretical Computer Science 721 (2018) 42–53]. This paper is dedicated to the precise calculation of edge values within the resulting subgraphs, enhancing our comprehension of their structural properties and implications. In this paper, we focus on solving the MSP of [Formula: see text] for any [Formula: see text] and for odd [Formula: see text], [Formula: see text] and [Formula: see text], where [Formula: see text] is obtained by the Cartesian product of a complete graph with [Formula: see text] vertices with a removal of a cycle on [Formula: see text] vertices. It has been observed that the graph obtained by deleting a cycle of length [Formula: see text] from a complete graph [Formula: see text] is isomorphic to a circulant graph [Formula: see text], while the particular case [Formula: see text] has been previously studied in [A. Syeda and M. Rajesh, MinLA of [Formula: see text] and its optimal layout into certain trees, The Journal of Supercomputing (2023), doi:10.1007/s11227-023-05140-3]. Our work extends the understanding of the MSP to a broader class of graphs with similar properties and applications.

If T or T is an algebraically k-quasiclass A operator acting on an infinite di- mensional separable Hilbert space and F is an operator commuting with T, and there exists a positive integer n such that F n has a finite rank, then we prove that Weyl's theorem holds for f(T)+F for every f2 H( (T)), where H( (T)) denotes the set of all analytic functions in a neighborhood of (T). Moreover, if T is an algebraically k-quasiclass A operator, then -Weyl's theorem holds for f(T). Also, we prove that if T or T is an algebraically k-quasiclass A operator then both the Weyl spectrum and the approximate point spectrum of T obey the spectral mapping theorem for every f2 H( (T)). We begin with some standard notation on Fredholm theory. Throughout this paper letH be a separable complex Hilbert space with inner producth ;i . Let B(H) and K(H) denote respectively, theC -algebra of all bounded linear operators and the ideal of compact operators acting on H. If T 2 B(H), we shall write kerT and ranT for the null space and the range of T respectively. Also let (T ) =dim kerT, (T ) =dim kerT and let (T ), a(T ) denote the spectrum, approximate point spectrum of T, respectively. Let p = p(T ) be the ascent of T; i.e., the smallest nonnegative integer p such that kerT p = kerT p+1 . If such an integer does not exist, we put p(T ) =1. Analogously, let q = q(T ) be the descent of T; i.e., the smallest nonnegative integer q such that ran T q =ranT q+1 , and if such an integer does not exist, we put q(T ) =1. It is well known that if p(T ) and q(T ) are both finite then p(T ) = q(T ). Moreover, 0 < p( T ) = q( T ) <1 precisely when is a pole of the resolvent of T, see Heuser (20, Proposition 50.2). An operator T 2 B(H) is called Fredholm if ranT is

The Turán number is the maximum number of edges in an ‐free graph on vertices. Let be any tree. The odd‐ballooning of , denoted by , is a graph obtained by replacing each edge of with an odd cycle containing the edge, and all new vertices of the odd cycles are distinct. In this paper, we determine the exact value of for sufficiently large and being good, which generalizes all the known results on for being a star, due to Erdős, Füredi, Gould, and Gunderson (1995), Hou, Qiu, and Liu (2018), and Yuan (2018), and provides some counterexamples with chromatic number three to a conjecture of Keevash and Sudakov (2004), on the maximum number of edges not in any monochromatic copy of in a 2‐edge‐coloring of a complete graph of order .

Large 2 P3-free graphs with bounded degree

A simple 2-matching in a graph is a subgraph all of whose nodes have degree $1$ or $2$. A simple 2-matching is called $k$-restricted if every connected component has $&gt;k$ edges. We consider the problem of finding a $k$-restricted simple 2-matching with a maximum number of edges, which is a relaxation of the problem of finding a Hamilton cycle in a graph. Our main result is a min-max theorem for the maximum number of edges in a 1-restricted simple 2-matching. We prove this result constructively by presenting a polynomial time algorithm for finding a 1-restricted simple 2-matching with a maximum number of edges.

Fix integers $t \ge r \ge 2$ and an $r$-uniform hypergraph $F$. We prove that the maximum number of edges in a $t$-partite $r$-uniform hypergraph on $n$ vertices that contains no copy of $F$ is $c_{t, F}{n \choose r} +o(n^r)$, where $c_{t, F}$ can be determined by a finite computation. We explicitly define a sequence $F_1, F_2, \ldots$ of $r$-uniform hypergraphs, and prove that the maximum number of edges in a $t$-chromatic $r$-uniform hypergraph on $n$ vertices containing no copy of $F_i$ is $\alpha_{t,r,i}{n \choose r} +o(n^r)$, where $\alpha_{t,r,i}$ can be determined by a finite computation for each $i\ge 1$. In several cases, $\alpha_{t,r,i}$ is irrational. The main tool used in the proofs is the Lagrangian of a hypergraph.

A graph Ω(Λ, Γ) is considered super edge magic if there exists a bijective function ϕ:Λ(Ω)∪Γ(Ω)→ {1, 2, 3, …,|Λ(Ω)|+|Γ(Ω)|} such that ϕ(τ1)+(τ1τ2)+ ϕ(τ2) is a constant for every edge τ1τ2 ∈ Γ(Ω), and ϕ(Λ(Ω))= {1, 2, 3, …,|Λ(Ω)|}. Furthermore, the super edge magic deficiency of a graph Ω, denoted as μs (Ω), is either the minimum non-negative integer η such that Ω∪ηK1 is a super edge magic graph or +∞ if such an integer η does not exist. In this paper, we investigate the super edge magic deficiency of certain families of graphs.

A lambda colouring (or L(2, 1)-colouring) of a graph is an assignment of non-negative integers (with minimum assignment 0) to its vertices such that the adjacent vertices must receive integers at least two apart and the vertices at distance two must receive distinct integers. The lambda chromatic number (or the $$\lambda $$ number) of a graph G is the least non-negative integer among all the maximum assigned non-negative integer over all possible lambda colouring of the graph G. Here we have shown that every edge maximum graph with lambda chromatic number $$t\ge 5$$ , admits an equitable L(2, 1) colouring. Further, we have established a classification result, identifying all possible n-vertex graphs with lambda chromatic number $$t\ge 3$$ , where $$n\ge t+1$$ , which contain the maximum number of edges. Such classification provides a complete solution to a problem posed more than two decades ago in 1996 by John P. Georges and David W. Mauro.

Induced matchings in subcubic graphs without short cycles

In this paper we describe together with an overview about other results the main ideas of our polynomial time approximation schemes for the maximum weight independent set problem (selecting a set of disjoint disks in the plane of maximum total weight) in disk graphs and for the maximum bisection problem (finding a partition of the vertex set into two subsets of equal cardinality with maximum number of edges between the subsets) in unit-disk graphs.

Let G be a symmetric connected graph without loops. Denote by b(G) the maximum number of edges in a bipartite subgraph of G. Determination of b(G) is polynomial for planar graphs ([6], [8]); in general it is an NP-complete problem ([5]). Edwards in [1], [2] found some estimates of b(G) which give, in particular,for a connected graph G of n vertices and m edges, whereand ﹛x﹜ denotes the smallest integer ≧ x.We give an 0(V3) algorithm which for a given graph constructs a bipartite subgraph B with at least f(m, n) edges, yielding a short proof of Edwards’ result.Further, we consider similar methods for obtaining some estimates for a particular case of the satisfiability problem. Let Φ be a Boolean formula of variables x1, …, xn.

We study the probabilistic degree over reals of the OR function on $n$ variables. For an error parameter $\epsilon$ in (0,1/3), the $\epsilon$-error probabilistic degree of any Boolean function $f$ over reals is the smallest non-negative integer $d$ such that the following holds: there exists a distribution $D$ of polynomials entirely supported on polynomials of degree at most $d$ such that for all $z \in \{0,1\}^n$, we have $Pr_{P \sim D} [P(z) = f(z) ] \geq 1- \epsilon$. It is known from the works of Tarui ({Theoret. Comput. Sci.} 1993) and Beigel, Reingold, and Spielman ({ Proc. 6th CCC} 1991), that the $\epsilon$-error probabilistic degree of the OR function is at most $O(\log n.\log 1/\epsilon)$. Our first observation is that this can be improved to $O{\log {{n}\choose{\leq \log 1/\epsilon}}}$, which is better for small values of $\epsilon$. In all known constructions of probabilistic polynomials for the OR function (including the above improvement), the polynomials $P$ in the support of the distribution $D$ have the following special structure:$P = 1 - (1-L_1).(1-L_2)...(1-L_t)$, where each $L_i(x_1,..., x_n)$ is a linear form in the variables $x_1,...,x_n$, i.e., the polynomial $1-P(x_1,...,x_n)$ is a product of affine forms. We show that the $\epsilon$-error probabilistic degree of OR when restricted to polynomials of the above form is $\Omega ( \log a/\log^2 a )$ where $a = \log {{n}\choose{\leq \log 1/\epsilon}}$. Thus matching the above upper bound (up to poly-logarithmic factors).

A polynomial algorithm for the k-cluster problem on the interval graphs