Related papers: On Legendre Cordial Labeling of Complete Graphs

A graph $G$ is called \textit{super edge-magic} if there exists a bijective function $f$ from $V(G) \cup E(G)$ to $\{1, 2, \ldots, |V(G) \cup E(G)|\}$ such that $f(V(G)) = \{1, 2, \ldots, |V(G)|\}$ and $f(x) + f(xy) + f(y)$ is a constant…

We introduce the index i(v) = 1 - X(S(v)) for critical points of a locally injective function f on the vertex set V of a simple graph G=(V,E). Here S(v) = {w in E | (v,w) in E, f(w)-f(v)<0} is the subgraph of the unit sphere at v in G. It…

Let $G$ be a graph with vertex set $V(G)$ and edge set $E(G)$, and $f$ be a 0-1 labeling of $E(G)$ so that the absolute difference in the number of edges labeled 1 and 0 is no more than one. Call such a labeling $f$ \emph{edge-friendly}. We…

A $k$-star colouring of a graph $G$ is a function $f:V(G)\to\{0,1,\dots,k-1\}$ such that $f(u)\neq f(v)$ for every edge $uv$ of $G$, and every bicoloured connected subgraph of $G$ is a star. The star chromatic number of $G$, $\chi_s(G)$, is…

Perfect graphs form one of the distinguished classes of finite simple graphs. In 2006, Chudnovsky, Robertson, Seymour and Thomas proved that a graph is perfect if and only if it has no odd holes and no odd antiholes as induced subgraphs,…

Given a locally injective real function g on the vertex set V of a finite simple graph G=(V,E), we prove the Poincare-Hopf formula f_G(t) = 1+t sum_{x in V} f_{S_g(x)}(t), where S_g(x) = { y in S(x), g(y) less than g(x) } and f_G(t)=1+f_0 t…

An $L(2,1)$-labelling of a finite graph $\Gamma$ is a function that assigns integer values to the vertices $V(\Gamma)$ of $\Gamma$ (colouring of $V(\Gamma)$ by ${\mathbb{Z}}$) so that the absolute difference of two such values is at least…

An ordering of the vertices of a graph is \emph{connected} if every vertex (but the first) has a neighbor among its predecessors. The greedy colouring algorithm of a graph with a connected order consists in taking the vertices in order, and…

A magic labelling of a graph $G$ with magic sum $s$ is a labelling of the edges of $G$ by nonnegative integers such that for each vertex $v\in V$, the sum of labels of all edges incident to $v$ is equal to the same number $s$. Stanley gave…

Given a (proper) vertex coloring $f$ of a graph $G$, say $f\colon V(G)\to \mathbb{N}$, the difference edge labelling induced by $f$ is a function $h\colon E(G)\to \mathbb{N}$ defined as $h(uv)=|f(u)-f(v)|$ for every edge $uv$ of $G$. A…

Let p be an odd prime, let S be a finite set of primes q congruent to 1 mod p but not mod p^2 and let G_S be the Galois group of the maximal p-extension of Q un-ramified outside of S. If r is a continuous homomorphism of G_S into GL_2(Z_p)…

Generalizing the notion of odd-sum colorings, a $\mathbb{Z}$-labeling of a graph $G$ is called a closed coloring with remainder $k\mod n$ if the closed neighborhood label sum of each vertex is congruent to $k\mod n$. If such colorings…

XOR-magic graph labelings form a special subclass of group distance magic labelings. A simple connected graph of order $2^n$ is called an open (respectively, closed) XOR-magic graph of power $n$ if its vertices can be labeled bijectively…

Which graphs admit an integer value harmonic function which is injective and surjective onto $\Z$? Such a function, which we call harmonic labeling, is constructed when the graph is the $\Z^2$ square grid. It is shown that for any finite…

A graph G(V, E) is word-representable if there exists a word w over V such that distinct letters x and y alternate in w iff $xy \in E$. We introduce p-complete squares and p-complete square-free word-representable graphs. A word is…

Let $p$ be an odd prime. For $b,c\in\mathbb Z$, Sun introduced the determinant $$D_p(b,c)=\left|(i^2+bij+cj^2)^{p-2}\right|_{1\leqslant i,j \leqslant p-1},$$ and investigated the Legendre symbol $(\frac{D_p(b,c)}p)$. Recently Wu, She and Ni…

Let \pi(G) denote the set of prime divisors of the order of a finite group G. The prime graph of G is the graph with vertex set \pi(G) with edges {p,q} if and only if there exists an element of order pq in G. In this paper, we prove that a…

Let $\mathbb{N}_0$ denote the set of all non-negative integers and $\mathcal{P}(\mathbb{N}_0)$ be its power set. An integer additive set-labeling (IASL) of a graph $G$ is an injective function $f:V(G)\to \mathcal{P}(\mathbb{N}_0)$ such that…

We give regular models for modular curves associated with (normalizer of) split and non-split Cartan subgroups of ${\mathrm{GL}}_2 ({\mathbb F}_p )$ (for $p$ any prime, $p\ge 5$). We then compute the group of connected components of the…

A graph $G$ is called collapsible if for every even subset $R\subseteq V(G)$, there is a spanning connected subgraph $H$ of $G$ such that $R$ is the set of vertices of odd degree in $H$. A graph is the reduction of $G$ if it is obtained…