Related papers: Interval colorings of complete bipartite graphs an…

We study completely independent spanning trees (CIST), \textit{i.e.}, trees that are both edge-disjoint and internally vertex-disjoint, in split graphs. We establish a correspondence between the existence of CIST in a split graph and some…

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…

This is the first paper in a series whose goal is to give a polynomial time algorithm for the $4$-coloring problem and the $4$-precoloring extension problem restricted to the class of graphs with no induced six-vertex path, thus proving a…

The chromatic number of signed graphs is defined recently. The coloring and clique problem of interval graphs has been studied and polynomial time algorithms are established. Here we consider these problems for signed interval graphs and…

Motivated by a conjecture of Liang [Y.-C. Liang. {\em Anti-magic labeling of graphs}. PhD thesis, National Sun Yat-sen University, 2013.], we introduce a restricted path packing problem in bipartite graphs that we call a $\mathtt{V}$-free…

We investigate the distributions of the number of: (1) monochromatic complete subgraphs over edgewise 2-colorings of complete graphs; and (2) monochromatic arithmetic progressions over 2-colorings of intervals, as statistical Ramsey theory…

An \emph{equitable $(q, r)$-tree-coloring} of a graph $G$ is a $q$-coloring of $G$ such that the subgraph induced by each color class is a forest of maximum degree at most $r$ and the sizes of any two color classes differ by at most $1.$…

For a given graph $G$, the least integer $k\geq 2$ such that for every Abelian group $\mathcal{G}$ of order $k$ there exists a proper edge labeling $f:E(G)\rightarrow \mathcal{G}$ so that $\sum_{x\in N(u)}f(xu)\neq \sum_{x\in N(v)}f(xv)$…

We consider a problem proposed by Linial and Wilf to determine the structure of graphs that allows the maximum number of $q$-colorings among graphs with $n$ vertices and $m$ edges. Let $T_r(n)$ denote the Tur\'{a}n graph - the complete…

It has recently been pointed out in both of the works [C. Shanguan, Y. Zhang, and G. Ge, {\em IEEE Trans. Inform. Theory}, 64(8):5755-5766 (2018)] and [Q. Yan, X. Tang, Q. Chen, and M. Cheng, {\em IEEE Commun. Lett.}, 22(2):236-239 (2018)]…

A b-coloring of a graph is a proper coloring such that every color class contains a vertex that is adjacent to all other color classes. The b-chromatic number of a graph G, denoted by \chi_b(G), is the maximum number t such that G admits a…

In [A. V. Ivashchenko, Contractible transformations do not change the homology groups of graphs, Discrete Mathematics 126 (1) (1994) p 159,170], Ivashchenko started with the study of contractible graphs, he began with this because they have…

A coloring of a complete bipartite graph is shuffle-preserved if it is the case that assigning a color $c$ to edges $(u, v)$ and $(u', v')$ enforces the same color assignment for edges $(u, v')$ and $(u',v)$. (In words, the induced subgraph…

Let $G$ be a Class 1 graph with maximum degree $4$ and let $t\geq 5$ be an integer. We show that any proper $t$-edge coloring of $G$ can be transformed to any proper $4$-edge coloring of $G$ using only transformations on $2$-colored…

We consider unavoidable chromatic patterns in $2$-colorings of the edges of the complete graph. Several such problems are explored being a junction point between Ramsey theory, extremal graph theory (Tur\'an type problems), zero-sum Ramsey…

For an integer $q\ge 2$ and an even integer $d$, consider the graph obtained from a large complete $q$-ary tree by connecting with an edge any two vertices at distance exactly $d$ in the tree. This graph has clique number $q+1$, and the…

Motivated by the problem in [6], which studies the relative efficiency of propositional proof systems, 2-edge colorings of complete bipartite graphs are investigated. It is shown that if the edges of $G=K_{n,n}$ are colored with black and…

Kostochka and Woodall (2001) conjectured that the square of every graph has the same chromatic number and list chromatic number. In 2015 Kim and Park disproved this conjecture for non-bipartite and bipartite graphs. It was asked by several…

Let $K_{n,n}$ be the complete bipartite graph with $n$ vertices in each side. For each vertex draw uniformly at random a list of size $k$ from a base set $S$ of size $s=s(n)$. In this paper we estimate the asymptotic probability of the…

An edge labeling of a graph distinguishes neighbors by sets (multisets, resp.), if for any two adjacent vertices $u$ and $v$ the sets (multisets, resp.) of labels appearing on edges incident to $u$ and $v$ are different. In an analogous way…