We give sufficient conditions for a tripartite graph to have a spanning subgraph consisting of complete graphs of order 3. This is applied to bound the strong chromatic number of a graph.

This paper aims to establish theoretical foundations of graph product multilayer networks (GPMNs), a family of multilayer networks that can be obtained as a graph product of two or more factor networks. Cartesian, direct (tensor), and strong product operators are considered, and then generalized. We first describe mathematical relationships between GPMNs and their factor networks regarding thei...

A variation of graph coloring known as a t-tone k-coloring assigns a set of t colors to each vertex of a graph from the set {1, . . . , k}, where the sets of colors assigned to any two vertices distance d apart share fewer than d colors in common. The minimum integer k such that a graph G has a ttone k-coloring is known as the t-tone chromatic number. We study the 2-tone chromatic number in thr...

A variation of graph coloring known as a t-tone k-coloring assigns a set of t colors to each vertex of a graph from the set {1, . . . , k}, where the sets of colors assigned to any two vertices distance d apart share fewer than d colors in common. The minimum integer k such that a graph G has a ttone k-coloring is known as the t-tone chromatic number. We study the 2-tone chromatic number in thr...

let $g=(v,e)$ be a connected simple graph. a labeling $f:v to z_2$ induces two edge labelings $f^+, f^*: e to z_2$ defined by $f^+(xy) = f(x)+f(y)$ and $f^*(xy) = f(x)f(y)$ for each $xy in e$. for $i in z_2$, let $v_f(i) = |f^{-1}(i)|$, $e_{f^+}(i) = |(f^{+})^{-1}(i)|$ and $e_{f^*}(i) = |(f^*)^{-1}(i)|$. a labeling $f$ is called friendly if $|v_f(1)-v_f(0)| le 1$. for a friendly labeling $f$ of...

Roussel and Rubio proved a lemma which is essential in the proof of the Strong Perfect Graph Theorem. We give a new short proof of the main case of this lemma. In this note, we also give a short proof of Hayward’s decomposition theorem for weakly chordal graphs, relying on a Roussel–Rubio-type lemma. We recall how Roussel–Rubio-type lemmas yield very short proofs of the existence of even pairs ...

A longest sequence S of distinct vertices of a graph G such that each vertex of S dominates some vertex that is not dominated by its preceding vertices, is called a Grundy dominating sequence; the length of S is the Grundy domination number of G. In this paper we study the Grundy domination number in the four standard graph products: the Cartesian, the lexicographic, the direct, and the strong ...

Let G1 ⊗ G2 be the strong product of a k-extendable graph G1 and an l-extendable graph G2, and X an arbitrary set of vertices of G1⊗G2 with cardinality 2[(k + 1)(l + 1)/2]. We show that G1 ⊗ G2 − X contains a perfect matching. It implies that G1 ⊗ G2 is [(k + 1)(l + 1)/2]-extendable, whereas the Cartesian product of G1 and G2 is only (k+l+1)extendable. As in the case of the Cartesian product, t...

In 1960 Berge came up with the concept of perfect graphs, and in doing so, conjectured some characteristics about them. A perfect graph is a graph in which the chromatic number of every induced subgraph equals the size of the largest clique of that subgraph [2]. Two conjectures are now known as the Perfect Graph Theorem and the Strong Perfect Graph Theorem. Both of these theorems make deteminin...