Colton Magnant

• Doctor of Philosophy
• Lead Data Scientist at United Parcel Service

Given a graph G and a positive integer k, define the Gallai—Ramsey number to be the minimum number of vertices n such that any k-edge coloring of Kn contains either a rainbow (all different colored) triangle or a monochromatic copy of G. In this paper, we obtain exact values of the Gallai—Ramsey numbers for the union of two stars in many cases and...

Given a graph G and a positive integer k, define the Gallai-Ramsey number to be the minimum number of vertices n such that any k-edge coloring of Kn\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargi...

Given a graph H $H$, the k $k$‐colored Gallai–Ramsey number g r k ( K 3 : H ) $g{r}_{k}({K}_{3}:H)$ is defined to be the minimum integer n $n$ such that every k $k$‐coloring of the edges of the complete graph on n $n$ vertices contains either a rainbow triangle or a monochromatic copy of H . $H.$ Fox et al. conjectured the values of the Gallai–Rams...

We show that if $\mathfrak{g}$ is a Frobenius seaweed, then the spectrum of the adjoint of a principal element consists of an unbroken set of integers whose multiplicities have a symmetric distribution. Our methods are combinatorial.

Given a graph H, the k-colored Gallai-Ramsey number grk(K3:H) is defined to be the minimum integer n such that every k-coloring of the edges of the complete graph on n vertices contains either a rainbow triangle or a monochromatic copy of H. Fox et al. [J. Fox, A. Grinshpun, and J. Pach. The Erdős-Hajnal conjecture for rainbow triangles. J. Combin....

Let G be an edge-colored connected graph. A path P in G is called \(\ell \)-rainbow if no two edges of the same color have fewer than \(\ell \) edges between them on P. The graph G is called \((k,\ell )\)-rainbow connected if every pair of distinct vertices of G are connected by k pairwise internally vertex-disjoint \(\ell \)-rainbow paths in G. Fo...

An edge-colored graph G is rainbow if each edge receives a different color. For a connected graph H, G is rainbow H-free if G does not contain a rainbow subgraph which is isomorphic to H. By definition, if H′ is a connected subgraph of H, every rainbow H′-free graph is rainbow H-free. In this note, we consider a kind of reverse implication, where H...

Given a graph G and a positive integer k, define the Gallai-Ramsey number to be the minimum number of vertices n such that any k-edge coloring of Kn\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargi...

When many colors appear in edge-colored graphs, it is only natural to expect rainbow subgraphs to appear. This anti-Ramsey problem has been studied thoroughly and yet there remain many gaps in the literature. Expanding upon classical and recent results forcing rainbow triangles to appear, we consider similar conditions which force the existence of...

Given a graph G and a positive integer k, define the Gallai–Ramsey number to be the minimum number of vertices n such that any k-edge coloring of Kn contains either a rainbow (all different colored) triangle or a monochromatic copy of G. In this paper, we obtain general upper and lower bounds on the Gallai–Ramsey numbers for the graph G=Str obtaine...

Given two disjoint sets of k vertices each in a graph G, it was recently asked whether δ(G)≥n/2\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\delta (G) \ge n/2$$\end{...

Given a graph $G$ and a positive integer $k$, define the \emph{Gallai-Ramsey number} to be the minimum number of vertices $n$ such that any $k$-edge coloring of the complete graph $K_n$ contains either a rainbow (all different colored) triangle or a monochromatic copy of $G$. In this paper, we obtain the exact value of the Gallai-Ramsey numbers for...

The general structure of colored complete graphs containing no copy of a particular rainbow subgraph has been extremely useful in establishing sharp Ramsey-type results for finding monochromatic subgraphs. Several small graphs, like \(P_{3}\) for example, immediately trivialize the problem. Indeed, if a colored complete graph contains no rainbow co...

Given graphs G and H and a positive integer k, the Gallai–Ramsey number, denoted by \(gr_{k}(G : H)\) is defined to be the minimum integer n such that every coloring of \(K_{n}\) using at most k colors will contain either a rainbow copy of G or a monochromatic copy of H. We consider this question in the cases where \(G \in \{P_{4}, P_{5}\}\). In th...

Recall that a Gallai coloring of is a rainbow triangle-free edge-coloring of . For a connected bipartite graph H, define s(H) to be the order of the smallest part in the unique bipartition of H and let be the order of the largest part.

Recall that is the star on 4 vertices with the addition of an extra edge between two of the leaves. This can also be seen as a triangle with an extra pendant edge. Fujita and Magnant proved the following result.

A graphgraph is a collection of vertices and a collection of (unordered) pairs of vertices called edges . Since each edge is a pair of vertices, we call each vertex u or v in e an end of the edge e. The two vertices at either end of an edge are adjacentadjacent. A graph is called simplesimple if every pair of vertices is contained in at most one ed...

This chapter concludes the book by summarizing the known sharp results to the best of our knowledge with appropriate citations. We also include a section with conjectures and open problems for those interested in pursuing future work in the area. These problems are by no means exhaustive since any unknown sharp value of a Gallai-Ramsey number would...

We show sharp Vizing-type inequalities for eternal domination. Namely, we prove that for any graphs G and H, where is the eternal domination function, α is the independence number, and is the strong product of graphs. This addresses a question of Klostermeyer and Mynhardt. We also show some families of graphs attaining the strict inequality where i...

Given graphs G and H and a positive integer k, the Gallai-Ramsey number is the minimum integer N such that for any integer every k-edge-coloring of Kn contains either a rainbow copy of G or a monochromatic copy of H. These numbers have recently been studied for the case when where still only a few precise numbers are known for all k. In this paper,...

Given a graph $G$ and a positive integer $k$, the \emph{Gallai-Ramsey number} is defined to be the minimum number of vertices $n$ such that any $k$-edge coloring of $K_n$ contains either a rainbow (all different colored) copy of $G$ or a monochromatic copy of $G$. In this paper, we obtain general upper and lower bounds on the Gallai-Ramsey numbers...

This book explores topics in Gallai-Ramsey theory, which looks into whether rainbow colored subgraphs or monochromatic subgraphs exist in a sufficiently large edge-colored complete graphs. A comprehensive survey of all known results with complete references is provided for common proof methods. Fundamental definitions and preliminary results with i...

An path that is edge-colored is called proper if no two consecutive edges receive the same color. A general graph that is edge-colored is called properly connected if, for every pair of vertices in the graph, there exists a properly colored path from one to the other. Given two vertices u and v in a properly connected graph G, the proper distance i...

Given a graph H, the k‐colored Gallai‐Ramsey number grk(K3⁢ :H) is defined to be the minimum integer n such that every k‐coloring (using all k colors) of the complete graph on n vertices contains either a rainbow triangle or a monochromatic copy of H. Recently, Fox et al [J. Combin. Theory Ser. B, 111 (2015), pp. 75–125] conjectured the value of th...

The study of graph Ramsey numbers within restricted colorings, in particular forbidding a rainbow triangle, has recently been blossoming under the name Gallai-Ramsey numbers. In this work, we extend the main structural tool from rainbow triangle free colorings of complete graphs to rainbow Berge triangle free colorings of hypergraphs. In doing so,...

Writing assignments in any mathematics course always present several challenges, particularly in lower-level classes where the students are not expecting to write more than a few words at a time. Developed based on strategies from several sources, the two small writing assignments included in this paper represent a gentle introduction to the writin...

Given a graph $G$ and a positive integer $k$, define the \emph{Gallai-Ramsey number} to be the minimum number of vertices $n$ such that any $k$-edge coloring of $K_n$ contains either a rainbow (all different colored) triangle or a monochromatic copy of $G$. In this paper, we obtain general upper and lower bounds on the Gallai-Ramsey numbers for the...

A recent variation of the classical geodetic problem, the strong geodetic problem, is defined as follows. If G is a graph, then \(\mathrm{sg}(G)\) is the cardinality of a smallest vertex subset S, such that one can assign a fixed geodesic to each pair \(\{x,y\}\subseteq S\) so that these \({|S|\atopwithdelims ()2}\) geodesics cover all the vertices...

Analogous to the Type-$A_{n-1}=\mathfrak{sl}(n)$ case, we show that if $\mathfrak{g}$ is a Frobenius seaweed subalgebra of $B_{n}=\mathfrak{so}(2n+1)$ or $C_{n}=\mathfrak{sp}(2n)$, then the spectrum of the adjoint of a principal element consists of an unbroken set of integers whose multiplicities have a symmetric distribution.

Analogous to the Type-A n−1 = sl(n) case, we show that if g is a Frobenius seaweed subalgebra of B n = so(2n + 1) or C n = sp(2n), then the spectrum of the adjoint of a principal element consists of an unbroken set of integers whose multiplicities have a symmetric distribution. Mathematics Subject Classification 2010 : 17B20, 05E15

Given a graph $G$ and a positive integer $k$, define the \emph{Gallai-Ramsey number} to be the minimum number of vertices $n$ such that any $k$-edge coloring of $K_n$ contains either a rainbow (all different colored) triangle or a monochromatic copy of $G$. Much like graph Ramsey numbers, Gallai-Ramsey numbers have gained a reputation as being very...

A colored complete graph is said to be Gallai-colored if it contains no rainbow triangle. This property has been shown to be equivalent to the existence of a partition of the vertices (of every induced subgraph) in which at most two colors appear on edges between the parts and at most one color appears on edges in between each pair of parts. We ext...

When many colors appear in edge-colored graphs, it is only natural to expect rainbow subgraphs to appear. This anti-Ramsey problem has been studied thoroughly and yet there remain many gaps in the literature. Expanding upon classical and recent results forcing rainbow triangles to appear, we consider similar conditions which force the existence of...

Writing assignments in any mathematics course always present several challenges, particularly in lower-level classes where the students are not expecting to write more than a few words at a time. Developed based on strategies from several sources, the two small writing assignments included in this paper represent a gentle introduction to the writin...

Given graphs $G$ and $H$ and a positive integer $k$, the Gallai-Ramsey number $gr_{k}(G : H)$ is the minimum integer $N$ such that for any integer $n \geq N$, every $k$-edge-coloring of $K_{n}$ contains either a rainbow copy of $G$ or a monochromatic copy of $H$. These numbers have recently been studied for the case when $G = K_{3}$, where still on...

Given a graph G and a positive integer k, the Gallai–Ramsey number is defined to be the minimum number of vertices n such that any k-edge coloring of K n contains either a rainbow (all different colored) triangle or a monochromatic copy of G. In this paper, we obtain general upper and lower bounds on the Gallai–Ramsey numbers for books B m =K 2 +K...

Given a graph G and a positive integer k, define the Gallai-Ramsey number to be the minimum number of vertices n such that any k-edge coloring of Kn contains either a rainbow (all different colored) triangle or a monochromatic copy of G. In this paper, we obtain general upper and lower bounds on the Gallai-Ramsey numbers for fans Fm = K1 + mK2 and...

Given graphs G and H and a positive integer k, the Gallai-Ramsey number, denoted by gr k (G : H) is defined to be the minimum integer n such that every coloring of K n using at most k colors will contain either a rainbow copy of G or a monochromatic copy of H. We consider this question in the cases where G ∈ {P 4 , P 5 }. In the case where G = P 4...

Given a graph H, the k-colored Gallai Ramsey number gr k (K3 : H) is defined to be the minimum integer n such that every k-coloring of the edges of the complete graph on n vertices contains either a rainbow triangle or a monochromatic copy of H. Fox et al. [J. Fox, A. Grinshpun, and J. Pach. The Erd˝ os-Hajnal conjecture for rainbow triangles. J. C...

A path in an edge-colored graph G is called a rainbow path if no two edges on the path have the same color. The graph G is called rainbow connected if between every pair of distinct vertices of G, there is a rainbow path. Recently, Johnson et al. considered this concept with the additional requirement that the coloring of G is proper. The proper ra...

A function f:V(G)→2[k] is an independent k-rainbow dominating function of a graph G if {x∣f(x)≠0̸} is an independent set of G and for every vertex x with f(x) = 0̸, we have [Formula presented]. The independent k-rainbow domination number of G, denoted irk(G), is the minimum, over all independent k-rainbow dominating functions f, ∑x∈V(G)|f(x)|. We p...

In 2000, Enomoto and Ota conjectured that if a graph $G$ satisfies $\sigma_{2}(G) \geq n + k - 1$, then for any set of $k$ vertices $v_{1}, \dots, v_{k}$ and for any positive integers $n_{1}, \dots, n_{k}$ with $\sum n_{i} = |G|$, there exists a partition of $V(G)$ into $k$ paths $P_{1}, \dots, P_{k}$ such that $v_{i}$ is an end of $P_{i}$ and $|P_...

In this paper, we study (zero) forcing sets which induce connected subgraphs of a graph. The minimum cardinality of such a set is called the connected forcing number of the graph. We provide sharp upper and lower bounds on the connected forcing number in terms of the minimum degree, maximum degree, girth, and order of the graph.

Let k be a positive integer and let F and \(H_{1}, H_{2}, \ldots , H_{k}\) be simple graphs. The proper-Ramsey number \(pr_{k}(F; H_{1}, H_{2}, \ldots , H_{k})\) is the minimum integer n such that any k-coloring of the edges of \(K_{n}\) contains either a properly colored copy of F or a copy of \(H_{i}\) in color i, for some i. We consider the case...

Given a fixed graph H, we provide a sharp degree-sum condition such that if a graph G is sufficiently large with sufficiently large minimum degree sum, then it contains a spanning H-subdivision in which the ground vertices and lengths of the corresponding edge paths are specified.

Gallai-Ramsey numbers often consider edge-colorings of complete graphs in which there are no rainbow triangles. Within such colored complete graphs, the goal is to look for specified monochromatic subgraphs. We consider an “off diagonal” case of this concept by looking for either a monochromatic K3\documentclass[12pt]{minimal} \usepackage{amsmath}...

Given a graph G and a positive integer k, define the Gallai-Ramsey number to be the minimum number of vertices n such that any k-edge coloring of Kn contains either a rainbow (all different colored) triangle or a monochromatic copy of G. In this paper, we consider two classes of unicyclic graphs, the star with an extra edge and the path with a tria...

Given a graph $G$ and a positive integer $k$, define the \emph{Gallai-Ramsey number} to be the minimum number of vertices $n$ such that any $k$-edge coloring of $K_n$ contains either a rainbow (all different colored) triangle or a monochromatic copy of $G$. In this paper, we consider two classes of unicyclic graphs, the star with an extra edge and...

If $\mathfrak{g}$ is a Frobenius Lie algebra, then for certain $F\in \mathfrak{g}^*$ the natural map $\mathfrak{g}\longrightarrow \mathfrak{g}^* $ given by $x \longmapsto F[x,-]$ is an isomorphism. The inverse image of $F$ under this isomorphism is called a principal element. We show that if $\mathfrak{g}$ is a Frobenius seaweed subalgebra of $A_{n...

The study of graph Ramsey numbers within restricted colorings, in particular forbidding a rainbow triangle, has recently been blossoming under the name Gallai-Ramsey numbers. In this work, we extend the main structural tool from rainbow triangle free colorings of complete graphs to rainbow Berge triangle free colorings of hypergraphs. In doing so,...

Given two graphs G and H and a positive integer k, the k-color Gallai-Ramsey number, denoted by gr k (G : H), is the minimum integer N such that for all n ≥ N , every k-coloring of the edges of K n contains either a rainbow copy of G or a monochromatic copy of H. We prove that gr k (K 3 : C 2ℓ+1) = ℓ · 2 k + 1 for all k ≥ 1 and ℓ ≥ 3.

First of all, we introduce some basis notions about computational complexity.

Using a minimum degree assumption to provide density, the following was shown for pc2(G). The proper 2-connection number pc2(G) is the minimum number of colors needed to color the edges of G so that between every pair of vertices, there are at least two internally disjoint proper paths. First we present an easy lemma without proof.

Among the many interesting problems of determining the proper connection numbers of graphs, it is worthwhile to study the proper connection number of G according to some constraints of the complementary graph.

For random graphs, the following results were shown in Gu et al. (Theor Comput Sci 609:336–343, 2016). Here let G(n, p) denote the Erdős-Renyi (Erdős and Rényi, Magy Tud Akad Mat Kutató Int Közl 5:17–61, 1960) random graph with n vertices and edges appearing with probability p. We say an event \(\mathcal {A}\) happens with high probability if the p...

In this chapter, we state some general results for proper connection number of graphs. There is an easy lower bound on pc(G) using the maximum number of bridges (cut edges) incident to a single vertex. All such bridges must receive distinct colors for the coloring to be proper connected, so the following result comes at no surprise.

Relating the proper connection number with domination, the following results were proven in Li et al. (Theor Comput Sci 607:480–487, 2015). A dominating set for a graph G = (V, E) is a subset D of V such that every vertex not in D is adjacent to at least one member of D. A dominating set D is called two-way two-step dominating if every pendant vert...

In this chapter, we provide several results concerning the proper connection number using connectivity to measure the number of edges and their distribution.

There have been several generalizations or extensions of the proper connection number. We discuss a few of these in this chapter.

Much like the undirected version, a strongly connected directed graph is called proper connected if between every ordered pair of vertices, there is a directed properly colored path. Defined in Magnant et al. (Matematiqki Vesnik 68(1):58–65, 2016), the directed proper connection number of a strongly connected directed graph G, denoted by \(\overrig...

In this chapter, we consider results which use assumptions on the degrees or number of edges, thereby driving the proper connection number down.

Notions of vertex proper connection, the vertex-coloring version of the proper connection number, have been defined and studied independently in Chizmar et al. (AKCE Int J Graphs Comb 13(2):103–106, 2016) and Jiang et al. (Bull Malays Math Sci Soc 41(1):415–425, 2018). A vertex-colored graph G is called proper vertex k-connected if every pair of ve...

Given a graph H and a positive integer k, the k-color Gallai-Ramsey number gr k (K 3 : H) is defined to be the minimum number of vertices n for which any k-coloring of the complete graph K n contains either a rainbow triangle or a monochromatic copy of H. The behavior of these numbers is rather well understood when H is bipartite but when H is not...

Given a graph G and a positive integer k, the Gallai-Ramsey number is defined to be the minimum number of vertices n such that any k-edge coloring of Kn contains either a rainbow (all different colored) triangle or a monochromatic copy of G. In this paper, we obtain general upper and lower bounds on the Gallai-Ramsey numbers for books Bm = K2 + Km...

Given a graph $G$ and a positive integer $k$, the \emph{Gallai-Ramsey number} is defined to be the minimum number of vertices $n$ such that any $k$-edge coloring of $K_n$ contains either a rainbow (all different colored) triangle or a monochromatic copy of $G$. In this paper, we obtain general upper and lower bounds on the Gallai-Ramsey numbers for...

A graph is said to be \emph{total-colored} if all the edges and the vertices of the graph are colored. The total-colored graph is \emph{total-rainbow connected} if any two vertices of the graph are connected by a path whose edges and internal vertices have distinct colors. For a connected graph $G$, the \emph{total-rainbow connection number} of $G$...

An edge-colored path is called properly colored if no two consecutive edges have the same color. An edge-colored graph is called properly connected if, between every pair of vertices, there is a properly colored path. Moreover, the proper distance between vertices u and v is the length of the shortest properly colored path from u to v. Given a part...

Gallai-colorings are edge-colored complete graphs in which there are no rainbow triangles. Within such colored complete graphs, we consider Ramsey-type questions, looking for specified monochromatic graphs. In this work, we consider monochromatic bipartite graphs since the numbers are known to grow more slowly than for non-bipartite graphs. The mai...

Gallai-colorings are edge-colored complete graphs in which there are no rainbow triangles. Within such colored complete graphs, we consider Ramsey-type questions, looking for specified monochromatic graphs. In this work, we consider monochromatic bipartite graphs since the numbers are known to grow more slowly than for non-bipartite graphs. The mai...

Let G be an edge-colored connected graph. A path P in G is called ℓ-rainbow if each subpath of length at most ℓ + 1 is rainbow. The graph G is called (k, ℓ)-rainbow connected if there is an edge-coloring such that every pair of distinct vertices of G is connected by k pairwise internally vertex-disjoint ℓ-rainbow paths in G. The minimum number of c...

We consider the connected graphs G that satisfy the following property: If (Formula presented.) are integers, then any coloring of the edges of (Formula presented.), using m colors, containing no properly colored copy of G, contains a monochromatic k-connected subgraph of order at least (Formula presented.) where f does not depend on n. If we let (...

As one of the counting-based topological indices, the number of subtrees and its variations has received much attention in recent years. In this paper, using generating functions, we investigate and derive formulas for this index of hexagonal and phenylene chains. We also present graph-theoretical algorithms for enumerating subtrees of these two ch...

A graph is said to be \emph{total-colored} if all the edges and the vertices of the graph are colored. A total-colored graph is \emph{total-rainbow connected} if any two vertices of the graph are connected by a path whose edges and internal vertices have distinct colors. For a connected graph $G$, the \emph{total-rainbow connection number} of $G$,...

Sharp minimum degree and degree sum conditions are proven for the existence of a Hamiltonian cycle passing through specified vertices with prescribed distances between them in large graphs.

An edge-coloured path is rainbow if its edges have distinct colours. For a connected graph $G$, the rainbow connection number (resp. strong rainbow connection number) of $G$ is the minimum number of colours required to colour the edges of $G$ so that, any two vertices of $G$ are connected by a rainbow path (resp. rainbow geodesic). These two graph...

An edge-coloured path is rainbow if its edges have distinct colours. For a connected graph $G$, the rainbow connection number (resp. strong rainbow connection number) of $G$ is the minimum number of colours required to colour the edges of $G$ so that, any two vertices of $G$ are connected by a rainbow path (resp. rainbow geodesic). These two graph...

We consider edge-colorings of complete graphs in which each color induces a subgraph that does not contain an induced copy of K1,t for some t > 3. It turns out that such colorings, if the underlying graph is sufficiently large, contain spanning monochromatic κ-connected subgraphs. Furthermore, there exists a color, say blue, such that every vertex...

Let $G$ be an edge-colored connected graph. A path $P$ in $G$ is called a distance $\ell$-proper path if no two edges of the same color can appear with less than $\ell$ edges in between on $P$. The graph $G$ is called $(k,\ell)$-proper connected if there is an edge-coloring such that every pair of distinct vertices of $G$ are connected by $k$ pairw...

Let $G$ be an edge-colored connected graph. A path $P$ in $G$ is called a distance $\ell$-proper path if no two edges of the same color appear with fewer than $\ell$ edges in between on $P$. The graph $G$ is called $(k,\ell)$-proper connected if every pair of distinct vertices of $G$ are connected by $k$ pairwise internally vertex-disjoint distance...

If $\mathfrak{g}$ is a Frobenius Lie algebra, then for certain $F\in \mathfrak{g}^*$ the natural map $\mathfrak{g}\longrightarrow \mathfrak{g}^* $ given by $x \longmapsto F[x,-]$ is an isomorphism. The inverse image of $F$ under this isomorphism is called a principal element. We show that if $\mathfrak{g}$ is a Frobenius seaweed subalgebra of $A_{n...

A tree $T$ in an edge-colored graph is called a {\it proper tree} if no two adjacent edges of $T$ receive the same color. Let $G$ be a connected graph of order $n$ and $k$ be an integer with $2\leq k \leq n$. For $S\subseteq V(G)$ and $|S| \ge 2$, an $S$-tree is a tree containing the vertices of $S$ in $G$. Suppose $\{T_1,T_2,\ldots,T_\ell\}$ is a...

A tree $T$ in an edge-colored graph is called a {\it proper tree} if no two adjacent edges of $T$ receive the same color. Let $G$ be a connected graph of order $n$ and $k$ be an integer with $2\leq k \leq n$. For $S\subseteq V(G)$ and $|S| \ge 2$, an $S$-tree is a tree containing the vertices of $S$ in $G$. Suppose $\{T_1,T_2,\ldots,T_\ell\}$ is a...

A path in an edge-colored graph is called a proper path if no two adjacent edges of the path receive the same color. For a connected graph G, the proper connection number of G is defined as the minimum number of colors needed to color its edges, so that every pair of distinct vertices of G is connected by at least one proper path in G. Recently, Li...

This note introduces the vertex proper connection number of a graph and provides a relationship to the chromatic number of minimally connected subgraphs. Also a notion of total proper connection is introduced and a question is asked about a possible relationship between the total proper connection number and the vertex and edge proper connection nu...

In this paper, we study (zero) forcing sets which induce connected subgraphs of a graph. The minimum cardinality of such a set is called the connected forcing number of the graph. We provide sharp upper and lower bounds on the connected forcing number in terms of the minimum degree, maximum degree, girth, and order of the graph.

We prove a sharp connectivity and degree sum condition for the existence of a subdivision of a multigraph in which some of the vertices are specified and the distance between each pair of vertices in the subdivision is prescribed (within one). Our proof makes use of the powerful Regularity Lemma in an easy way that highlights the extreme versatilit...

A path in an edge-colored graph is called proper if no two consecutive edges of the path receive the same color. For a connected graph $G$, the proper connection number $pc(G)$ of $G$ is defined as the minimum number of colors needed to color its edges so that every pair of distinct vertices of $G$ are connected by at least one proper path in $G$....

A path in an edge-colored graph is called proper if no two consecutive edges of the path receive the same color. For a connected graph $G$, the proper connection number $pc(G)$ of $G$ is defined as the minimum number of colors needed to color its edges so that every pair of distinct vertices of $G$ are connected by at least one proper path in $G$....

Analogous to the sl(n) case, we address the computation of the index of seaweed subalgebras of sp(2n) by introducing graphical representations called symplectic meanders. Formulas for the algebra's index may be computed by counting the connected components of its associated meander. In certain cases, formulas for the index can be given in terms of...

Analogous to the sl(n) case, we address the computation of the index of seaweed subalgebras of sp(2n) by introducing graphical representations called symplectic meanders. Formulas for the algebra's index may be computed by counting the connected components of its associated meander. In certain cases, formulas for the index can be given in terms of...

A path in an edge-colored graph is called a proper path if no two adjacent edges of the path receive the same color. For a connected graph $G$, the proper connection number $pc(G)$ of $G$ is defined as the minimum number of colors needed to color its edges, so that every pair of distinct vertices of $G$ is connected by at least one proper path in $...

The path partition number of a graph is the minimum number of paths required to partition the vertices. We consider upper bounds on the path partition number under minimum and maximum degree assumptions.

The concept of graceful labeling of graphs has been extensively studied. In 1994, Mitchem and Simoson introduced a stronger concept called super-edge-graceful labeling for some classes of graphs. Among many other interesting pioneering results, Mitchem and Simoson provided a simple but powerful recursive way of constructing super-edge-graceful tree...

An edge-colored directed graph is called properly connected if, between every pair of vertices, there is a properly colored directed path. We study some conditions on directed graphs which guarantee the existence of a coloring that is properly connected. We also study conditions on a colored directed graph which guarantee that the coloring is prope...

Given a graph G, the atom bond connectivity (ABC) index is defined to be ABC(G) = Sigma(u similar to v) root d(u)+d(v)-2/d(u)d(v) where u and v are vertices of G, d(u) denotes the degree of the vertex u, and u similar to v indicates that u and v are adjacent. Although it is known that among trees of a given order n, the star has maximum ABC index,...

An edge-coloured path is rainbow if the colours of its edges are distinct. For a positive integer k, an edge-colouring of a graph G is rainbow k-connected if any two vertices of G are connected by k internally vertex-disjoint rainbow paths. The rainbow k-connection number rc k (G) is defined to be the minimum integer t such that there exists an edg...

Sheehan conjectured in 1975 that every Hamiltonian regular simple graph of even degree at least four contains a second Hamiltonian cycle. We prove that most claw-free Hamiltonian graphs with minimum degree at least 3 have a second Hamiltonian cycle and describe the structure of those graphs not covered by our result. By this result, we show that Sh...