Proving a conjecture on the upper bound of semistrong chromatic indices of graphs

Let $G=(V(G), E(G))$ be a graph with maximum degree $\Delta$. For a subset $M$ of $E(G)$, we denote by $G[V(M)]$ the subgraph of $G$ induced by the endvertices of edges in $M$. We call $M$ a semistrong matching if each edge of $M$ is incident with a vertex that is of degree 1 in $G[V(M)]$. Given a positive integer $k$, a semistrong $k$-edge-coloring of $G$ is an edge coloring using at most $k$ colors in which each color class is a semistrong matching of $G$. The semistrong chromatic index of $G$, denoted by $\chi'_{ss}(G)$, is the minimum integer $k$ such that $G$ has a semistrong $k$-edge-coloring. Recently, Lu\v{z}ar, Mockov\v{c}iakov\'a and Sot\'ak conjectured that $\chi'_{ss}(G)\le \Delta^{2}-1$ for any connected graph $G$ except the complete bipartite graph $K_{\Delta,\Delta}$. In this paper, we settle this conjecture by proving that each such graph $G$ other than a cycle on $7$ vertices has a semistrong edge coloring using at most $\Delta^{2}-1$ colors.Comment: 20 pages, 9 figure

On n-fold L(j,k)-and circular L(j,k)-labelings of graphs

AbstractWe initiate research on the multiple distance 2 labeling of graphs in this paper.Let n,j,k be positive integers. An n-fold L(j,k)-labeling of a graph G is an assignment f of sets of nonnegative integers of order n to the vertices of G such that, for any two vertices u,v and any two integers a∈f(u), b∈f(v), |a−b|≥j if uv∈E(G), and |a−b|≥k if u and v are distance 2 apart. The span of f is the absolute difference between the maximum and minimum integers used by f. The n-fold L(j,k)-labeling number of G is the minimum span over all n-fold L(j,k)-labelings of G.Let n,j,k and m be positive integers. An n-fold circular m-L(j,k)-labeling of a graph G is an assignment f of subsets of {0,1,…,m−1} of order n to the vertices of G such that, for any two vertices u,v and any two integers a∈f(u), b∈f(v), min{|a−b|,m−|a−b|}≥j if uv∈E(G), and min{|a−b|,m−|a−b|}≥k if u and v are distance 2 apart. The minimum m such that G has an n-fold circular m-L(j,k)-labeling is called the n-fold circular L(j,k)-labeling number of G.We investigate the basic properties of n-fold L(j,k)-labelings and circular L(j,k)-labelings of graphs. The n-fold circular L(j,k)-labeling numbers of trees, and the hexagonal and p-dimensional square lattices are determined. The upper and lower bounds for the n-fold L(j,k)-labeling numbers of trees are obtained. In most cases, these bounds are attainable. In particular, when k=1 both the lower and the upper bounds are sharp. In many cases, the n-fold L(j,k)-labeling numbers of the hexagonal and p-dimensional square lattices are determined. In other cases, upper and lower bounds are provided. In particular, we obtain the exact values of the n-fold L(j,1)-labeling numbers of the hexagonal and p-dimensional square lattices

Circular chromatic numbers of some distance graphs

AbstractGiven a set D of positive integers, the distance graph G(Z,D) has vertices all integers Z, and two vertices j and j′ in Z are adjacent if and only if |j-j′|∈D. This paper determines the circular chromatic numbers of some distance graphs

Several parameters of generalized Mycielskians

AbstractThe generalized Mycielskians (also known as cones over graphs) are the natural generalization of the Mycielski graphs (which were first introduced by Mycielski in 1955). Given a graph G and any integer m⩾0, one can transform G into a new graph μm(G), the generalized Mycielskian of G. This paper investigates circular clique number, total domination number, open packing number, fractional open packing number, vertex cover number, determinant, spectrum, and biclique partition number of μm(G)

Study on fluorescence characteristics of the Ho 3+ :ZBLAN fiber under ~640 nm excitation

Abstract(#br)We investigated the absorption and emission characteristics of the Ho 3+ :ZBLAN fiber under ~640 nm excitation. Based on the Judd-Ofelt theory, a detailed spectroscopic analysis on excited states 5 I 6 and 5 I 7 of the Ho 3+ -ion was performed. The population dynamics was conducted by using the rate equation method, and a set of analytical expressions for population densities of various levels were obtained at steady state. Moreover, the fluorescence intensities of the 5 S 2 , 5 F 4 → 5 I 7 , 5 I 8 and 5 I 6 → 5 I 7 , 5 I 8 and 5 I 7 → 5 I 8 transitions were measured in different pumping powers. The simulated and experimental results are quite consistent. This work could provide the spectral information for optimal design of the visible oscillations in the Ho 3+ :ZBLAN fiber excited at ~640 nm

Multicoloring and Mycielski construction

AbstractThe generalized Mycielskians of graphs (also known as cones over graphs) are the natural generalization of the Mycielskians of graphs (which were first introduced by Mycielski in 1955). Given a graph G and any integer p⩾0, one can transform G into a new graph μp(G), the p-Mycielskian of G. In this paper, we study the kth chromatic numbers χk of Mycielskians and generalized Mycielskians of graphs. We show that χk(G)+1⩽χk(μ(G))⩽χk(G)+k, where both upper and lower bounds are attainable. We then investigate the kth chromatic number of Mycielskians of cycles and determine the kth chromatic number of p-Mycielskian of a complete graph Kn for any integers k⩾1, p⩾0 and n⩾2. Finally, we prove that if a graph G is a/b-colorable then the p-Mycielskian of G, μp(G), is (at+bp+1)/bt-colorable, where t=∑i=0p(a-b)ibp-i. And thus obtain graphs G with m(G) grows exponentially with the order of G, where m(G) is the minimal denominator of a a/b-coloring of G with χf(G)=a/b

Maximum weight t-sparse set problem on vector-weighted graphs

Let $t$ be a nonnegative integer and $G=(V(G),E(G))$ be a graph.     For $v \in V(G)$, let $N_{G}(v)=\{u \in V(G) \setminus \{v\}: uv \in E(G)\}$. And for $S\subseteq V(G)$, we define $d_{S}(G;v)= \left|N_{G}(v)\cap S \right|$ for $v\in S$ and $d_{S}(G;v)=-1$ for  $v\in V(G)\setminus S$. A subset $S\subseteq V(G)$  is called  a $t$-sparse set of  $G$ if  the maximum degree of the  induced subgraph $G[S]$  does not exceed  $t$.  In particular, a $0$-sparse set is precisely an independent set. A vector-weighted graph $(G,\vec{w},t)$ is a graph $G$ with a  vector weight function $\vec{w}: V(G) \to \mathbb{R}^{t+2}$, where $\vec{w}(v)=(w(v;-1),w(v;0),\ldots,w(v;t))$ for each $v\in V(G)$. The weight of  a $t$-sparse set $S$ in $(G,\vec{w},t)$ is  defined as  $\vec{w}(S,G)=\sum_{v\in V(G)} w(v;d_{S}(G;v))$. And a $t$-sparse set $S$ is a  maximum  weight $t$-sparse set of $(G,\vec{w},t)$ if there is no  $t$-sparse set of larger weight in $(G,\vec{w},t)$.  In this paper, we propose the   maximum  weight $t$-sparse set   problem on vector-weighted graphs, which is to find a  maximum  weight $t$-sparse set of  $(G,\vec{w},t)$. We  design a dynamic programming  algorithm to find a maximum  weight  $t$-sparse set of an  outerplane graph  $(G,\vec{w},t)$ which takes $O((t+2)^{4}n)$ time, where $n=\left|V(G)\right|$. Moreover, we give a polynomial-time algorithm for this problem on graphs with bounded treewidth

Some Results on the Erdős–Faber–Lovász Conjecture

Erdős–Faber–Lovász conjecture states that if a graph G is a union of the n edge-disjoint copies of complete graph Kn, that is, each pair of complete graphs has at most one shared vertex, then the chromatic number of graph G is n. In fact, we only need to consider the graphs where each pair of complete graphs has exactly one shared vertex. However, each shared vertex may be shared by more than two complete graphs. Therefore, this paper first considers the graphs where each shared vertex happens to be shared by two complete graphs, and then discusses the graphs with only one shared vertex shared by more than two complete graphs. The conjecture is correct for these two kinds of graphs in this work. Finally, the graph where each shared vertex happens to be shared by three complete graphs has been studied, and the conjecture also holds for such graphs when n=13. The graphs discussed in this paper have certain symmetric properties. The symmetry of graphs plays an important role in coloring. This work is an attempt to combine the symmetry of graphs with the coloring of graphs