Related papers: $L(p,q)$-Labeling of Graphs with Interval Represen…

A proper edge coloring of a graph $G$ with colors $1,2,\dots,t$ is called a \emph{cyclic interval $t$-coloring} if for each vertex $v$ of $G$ the edges incident to $v$ are colored by consecutive colors, under the condition that color $1$ is…

The covering radius of permutation group codes are studied in this paper with $l_{\infty}$-metric. We determine the covering radius of the $(p,q)$-type group, which is a direct product of two cyclic transitive groups. We also deduce the…

This paper gives tight upper bounds on the largest eigenvalue q(G) of the signless Laplacian of graphs with no 4-cycle and no 5-cycle. If n is odd, let F_{n} be the friendship graph of order n; if n is even, let F_{n} be F_{n-1} with an…

Several concepts that model processes of spreading (of information, disease, objects, etc.) in graphs or networks have been studied. In many contexts, we assume that some vertices of a graph $G$ are contaminated initially, before the…

A \emph{mixed interval graph} is an interval graph that has, for every pair of intersecting intervals, either an arc (directed arbitrarily) or an (undirected) edge. We are particularly interested in scenarios where edges and arcs are…

Given positive integers $p$ and $q$, a $(p,q)$-coloring of the complete graph $K_n$ is an edge-coloring in which every $p$-clique receives at least $q$ colors. Erd\H{o}s and Shelah posed the question of determining $f(n,p,q)$, the minimum…

We consider the problems of finding optimal identifying codes, (open) locating-dominating sets and resolving sets of an interval or a permutation graph. In these problems, one asks to find a subset of vertices, normally called a…

We find exact and asymptotic formulas for the number of pairs $(p,q)$ of $N$-cycles such that the all cycles of the product $p\cdot q$ have lengths from a given integer set. We then apply these results to prove a surprisingly high lower…

Let $ \Pi_q $ be the projective plane of order $ q $, let $\psi(m):=\psi(L(K_m))$ the pseudoachromatic number of the complete line graph of order $ m $, let $ a\in \{ 3,4,\dots,\tfrac{q}{2}+1 \} $ and $ m_a=(q+1)^2-a $. In this paper, we…

Upper bounds on the maximum number of minimal codewords in a binary code follow from the theory of matroids. Random coding provide lower bounds. In this paper we compare these bounds with analogous bounds for the cycle code of graphs. This…

An edge-coloring of a multigraph G with colors 1,2,...,t is called an interval t-coloring if all colors are used, and the colors of edges incident to any vertex of G are distinct and form an interval of integers. In this paper we prove that…

We give a sharp bound on the number of triangles in a graph with fixed number of edges. We also characterize graphs that achieve the maximum number of triangles. Using the upper bound on number of triangles, we prove that if $G$ is a…

We initiate the study of a new parameterization of graph problems. In a multiple interval representation of a graph, each vertex is associated to at least one interval of the real line, with an edge between two vertices if and only if an…

An $(r,s)$-directed hypergraph is a directed hypergraph with $r$ vertices in tail and $s$ vertices in head of each arc. Let $G$ be an $(r,s)$-directed hypergraph. For any real numbers $p$, $q\geq 1$, we define the $(p,q)$-spectral radius…

This paper gives a tight upper bound on the spectral radius of the signless Laplacian of graphs of given order and clique number. More precisely, let G be a graph of order n, let A be its adjacency matrix, and let D be the diagonal matrix…

A graph $G$ is called interval colorable if it has a proper edge coloring with colors $1,2,3,\dots$ such that the colors of the edges incident to every vertex of $G$ form an interval of integers. Not all graphs are interval colorable; in…

This paper studies the strong fractional choice number $ch^s_f(G)$ and the strong fractional paint number $\chi^s_{f,P}(G)$ of a graph $G$. We prove that these parameters of any finite graph are rational numbers. On the other hand, for any…

For $p,q\ge2$ the $\{p,q\}$-tiling graph is the (finite or infinite) planar graph $T_{p,q}$ where all faces are cycles of length $p$ and all vertices have degree $q$. We give algorithms for the problem of recognizing (induced) subgraphs of…

In this paper, we study the $L(2, 1)$-Labeling of the Mycielski and the iterated Mycielski of graphs in general. For a graph $G$ and all $t\geq 1$, we give sharp bounds for $\lambda(M^t(G))$ the $L(2, 1)$-labeling number of the $t$-th…

In this paper, we determine the achromatic and diachromatic numbers of some circulant graphs and digraphs each one with two lengths and give bounds for other circulant graphs and digraphs with two lengths. In particular, for the achromatic…