Related papers: From semi-total to equitable total colorings
Coloring a graph $G$ consists in finding an assignment of colors $c: V(G)\to\{1,\ldots,p\}$ such that any pair of adjacent vertices receives different colors. The minimum integer $p$ such that a coloring exists is called the chromatic…
Consider a simple graph $G=(V,E)$ of maximum degree $\Delta$ and its proper total colouring $c$ with the elements of the set $\{1,2,\ldots,k\}$. The colouring $c$ is said to be \emph{neighbour sum distinguishing} if for every pair of…
Let $r \geqslant 0$ and $k \geqslant 1$ be integers. We say that a graph $G$ has an $r$-equitable $k$-coloring if there exists a proper $k$-coloring of $G$ such that the sizes of any two color classes differ by at most $r$. The least $k$…
Let $G$ be a graph and c a proper k-coloring of G, i.e. any two adjacent vertices u and v have different colors c(u) and c(v). A proper k-coloring is a b-coloring if there exists a vertex in every color class that contains all the colors in…
A graph is said to be {\it total-colored} if all the edges and the vertices of the graph is colored. A path in a total-colored graph is a {\it total proper path} if $(i)$ any two adjacent edges on the path differ in color, $(ii)$ any two…
Hadwiger's conjecture is one of the most important and long-standing conjectures in graph theory. Reed and Seymour showed in 2004 that Hadwiger's conjecture is true for line graphs. We investigate this conjecture on the closely related…
A total coloring of a simple undirected graph $G$ is an assignment of colors to its vertices and edges such that the colors given to the vertices form a proper vertex coloring, the colors given to the edges form a proper edge coloring, and…
A proper coloring of vertices of a graph is equitable if the sizes of any two color classes differ by at most 1. Such colorings have many applications and are interesting by themselves. In this paper, we discuss the state of art and…
A proper coloring of a graph is \emph{conflict-free} if, for every non-isolated vertex, some color is used exactly once on its neighborhood. Caro, Petru\v{s}evski, and \v{S}krekovski proved that every graph $G$ has a proper conflict-free…
If $k\geq 0$, then a $k$-edge-coloring of a graph $G$ is an assignment of colors to edges of $G$ from the set of $k$ colors, so that adjacent edges receive different colors. A $k$-edge-colorable subgraph of $G$ is maximum if it is the…
We consider (not necessarily proper) colorings of the vertices of a graph where every color is thoroughly distributed, that is, appears in every open neighborhood. Equivalently, every color is a total dominating set. We define $\td(G)$ as…
With any (not necessarily proper) edge $k$-colouring $\gamma:E(G)\longrightarrow\{1,\dots,k\}$ of a graph $G$,one can associate a vertex colouring $\sigma\_{\gamma}$ given by $\sigma\_{\gamma}(v)=\sum\_{e\ni v}\gamma(e)$.A…
Let $2\le k\in\mathbb{Z}$. A total coloring of a $k$-regular simple graph via $k+1$ colors is an {\it efficient total coloring} if each color yields an efficient dominating set, where the efficient domination condition applies to the…
An edge-coloring of a graph $G$ with natural numbers is called a sum edge-coloring if the colors of edges incident to any vertex of $G$ are distinct and the sum of the colors of the edges of $G$ is minimum. The edge-chromatic sum of a graph…
A graph is 1-planar if it can be drawn on a plane so that each edge is crossed by at most one other edge. In this paper, we first give a useful structural theorem for 1-planar graphs, and then apply it to the list edge and list total…
Hadwiger Conjecture has been an open problem for over a half century1,6, which says that there is at most a complete graph Kt but no Kt+1 for every t-colorable graph. A few cases of Hadwiger Conjecture, such as 1, 2, 3, 4, 5, 6-colorable…
A total dominator coloring of a graph G is a proper coloring of G in which each vertex of the graph is adjacent to every vertex of some color class. The total dominator chromatic number of a graph is the minimum number of color classes in a…
A total coloring of a graph $G$ is a coloring of its vertices and edges such that no adjacent vertices, edges, and no incident vertices and edges obtain the same color. An interval total $t$-coloring of a graph $G$ is a total coloring of…
Let $G=(V,E)$ be a graph. A (proper) $k$-edge-coloring is a coloring of the edges of $G$ such that any pair of edges sharing an endpoint receive distinct colors. A classical result of Vizing ensures that any simple graph $G$ admits a…
Given a graph $G$, the total dominator coloring problem seeks a proper coloring of $G$ with the additional property that every vertex in the graph is adjacent to all vertices of a color class. We seek to minimize the number of color…