1-planar graphs are odd 13-colorable

The total chromatic number of a graph $G$, denoted by $\chi′′(G)$, is the minimum number of colors needed to color the vertices and edges of $G$ such that no two adjacent or incident elements get the same color. It is known that if a planar graph $G$ has maximum degree $\Delta ≥ 9$, then $\chi′′(G) = \Delta + 1$. The join $K_1 \vee P_n$ of $K_1$ and $P_n$ is called a fan graph $F_n$. In this paper, we prove that if $G$ is a $F_5$-free planar graph with maximum degree 8, then $\chi′′(G) = 9$.

Odd graph and its applications to the strong edge coloring

We introduce the notion of locally identifying coloring of a graph. A proper vertex-coloring $c$ of a graph $G$ is said to be locally identifying, if for any adjacent vertices $u$ and $v$ with distinct closed neighborhoods, the sets of colors that appear in the closed neighborhood of $u$ and $v$, respectively, are distinct. Let $\chi_{\rm{lid}}(G)$ be the minimum number of colors used in a locally identifying vertex-coloring of $G$. In this paper, we give several bounds on $\chi_{\rm{lid}}$ for different families of graphs (planar graphs, some subclasses of perfect graphs, graphs with bounded maximum degree) and prove that deciding whether $\chi_{\rm{lid}}(G)=3$ for a subcubic bipartite graph $G$ with large girth is an NP-complete problem.

The M‐degree of an edge xy in a graph is the maximum of the degrees of x and y. The M‐degree of a graph G is the minimum over M‐degrees of its edges. In order to get upper bounds on the game chromatic number, He et al showed that every planar graph G without leaves and 4‐cycles has M‐degree at most 8 and gave an example of such a graph with M‐degree 3. This yields upper bounds on the game chromatic number of C4‐free planar graphs. We determine the maximum possible M‐degrees for planar, projective‐planar and toroidal graphs without leaves and 4‐cycles. In particular, for planar and projective‐planar graphs this maximum is 7. © 2008 Wiley Periodicals, Inc. J Graph Theory 60: 80–85, 2009

The rainbow connection was first introduced by Chartrand in 2006 and then in 2009 Krivelevich and Yuster first time introduced the rainbow vertex connection. Let graph be a connected graph. The rainbow vertex-connection is the assignment of color to the vertices of a graph , if every vertex on the graph is connected by a path graph that has interior vertices in different colors. The minimum number of colors from the rainbow vertex coloring in the graph is called rainbow vertex connection number which is denoted . The results of the research are the rainbow vertex connection number of bull graph, net graph, triangular ladder graph, and graph composition (Pn[P1]).

The delunification process and minimal diagrams

A dominator coloring of a graph $G$ is a proper coloring in which every vertex of $G$ dominates at least one color class. The dominator chromatic number of $G, \chi_d(G)$, is defined by the minimum number of colors needed in a dominator coloring of $G$. A total dominator coloring of $G$ is a proper coloring of $G$ with the extra property that every vertex in $G$ properly dominates a color class. The total dominator chromatic number of $G, \chi_{t d}(G)$, is the minimum number of colors needed in a total dominator coloring of $G$. In this paper, we obtain dominator and total dominator chromatic number of Mongolian tent graphs and fire cracker graphs.

For a real number $r\ge 2$, a circular $r$-colouring of a signed graph $(G, \sigma)$ is a mapping $c: V(G)\to [0, r)$ such that $|c(x)-c(y)|\in [1,r-1]$ for each positive edge $xy$ and $|c(x)-c(y)|\in [0,r/2-1]\cup [r/2+1,r)$ for each negative edge $xy$. This concept is recently introduced by Naserasr, Wang, and Zhu in 2021, and they show that for any $\varepsilon>0$, there exist signed planar bipartite graphs (of girth 4) which are not circular $(4-\varepsilon)$-colourable. In this paper, we prove that for each signed planar graph $(G, \sigma)$ of girth at least $5$, there exists a real number $\varepsilon=\varepsilon(G,\sigma)>0$ such that $(G, \sigma)$ is circular $(4-\varepsilon)$-colorable. Our proof utilizes a Thomassen-type inductive argument on the dual version in terms of circular flows, which is motivated by a result of Richter, Thomassen, and Younger (2016) on group connectivity of $5$-edge-connected planar graphs.

An adjacent vertex distinguishing total coloring of a graph G is a proper total coloring of G such that any pair of adjacent vertices have distinct sets of colors. The minimum number of colors needed for an adjacent vertex distinguishing total coloring of G is denoted by $\chi''_{a}(G)$ . In this paper, we characterize completely the adjacent vertex distinguishing total chromatic number of planar graphs G with large maximum degree Δ by showing that if Δ?14, then $\varDelta+1\leq \chi''_{a}(G)\leq \varDelta+2$ , and $\chi''_{a}(G)=\varDelta+2$ if and only if G contains two adjacent vertices of maximum degree.

Let G be a plane graph. A vertex-colouring $$\varphi $$ of G is called facial non-repetitive if for no sequence $$r_1 r_2 \ldots r_{2n}$$ , $$n\ge 1$$ , of consecutive vertex colours of any facial path it holds $$r_i=r_{n+i}$$ for all $$i=1,2,\ldots ,n$$ . A plane graph G is facial non-repetitively k -choosable if for every list assignment $$L:V\rightarrow 2^{\mathbb {N}}$$ with minimum list size at least k there is a facial non-repetitive vertex-colouring $$\varphi $$ with colours from the associated lists. The facial Thue choice number, $$\pi _{fl}(G)$$ , of a plane graph G is the minimum number k such that G is facial non-repetitively k-choosable. We use the so-called entropy compression method to show that $$\pi _{fl} (G)\le c \varDelta $$ for some absolute constant c and G a plane graph with maximum degree $$\varDelta $$ . Moreover, we give some better (constant) upper bounds on $$\pi _{fl} (G)$$ for special classes of plane graphs.

Acyclic and k-distance coloring of the grid

The classical Nelson — Erdős — Hadwiger problem, which was proposed in the late 1940’s (see [1], [2]), consists in finding or at least estimating the so-called chromatic number χ(ℝ d ) of the Euclidean space ℝ d , where the value χ(ℝ d ) is defined as the minimum number of colours needed to paint all the points in ℝ d in such a way that any two points at the distance 1 apart receive different colours. Of course, in this definition, distance 1 may be replaced by an arbitrary fixed distance a > 0, due to the homogeneity of the Euclidean space. In other words, the quantity χ(ℝ d ) is the usual chromatic number of a graph \(\mathfrak{G}_a^d = \left( {{\mathfrak{G}^d},\mathfrak{E}_a^d} \right)\), provided $${\mathfrak{V}^d} = {\mathbb{R}^d},\mathfrak{E}_a^d = \{ (a,y) \in {\mathfrak{V}^d} \times {\mathfrak{V}^d}:|a - y| = a\} ,\,a > 0$$ .

A graph in this paper is nontrivial, finite, connected, simple, and undirected. Graph consists of a vertex set and edge set. Let u,v be two elements in vertex set, and q is the cardinality of edge set in G, a bijective function from the edge set to the first q natural number is called a vertex local antimagic edge labelling if for any two adjacent vertices and , the weight of is not equal with the weight of , where the weight of (denoted by ) is the sum of labels of edges that are incident to . Furthermore, any vertex local antimagic edge labelling induces a proper vertex colouring on where is the colour on the vertex . The vertex local antimagic chromatic number is the minimum number of colours taken over all colourings induced by vertex local antimagic edge labelling of . In this paper, we discuss about the vertex local antimagic chromatic number on disjoint union of some family graphs, namely path, cycle, star, and friendship, and also determine the lower bound of vertex local antimagic chromatic number of disjoint union graphs. The chromatic numbers of disjoint union graph in this paper attend the lower bound.

Let G be a connected and undirected graph. Vertex coloring in a graph G is a mapping from the set of vertices in G to the set of colors such that every two adjacent vertices have different colors. There are many types of vertex coloring, such as complete coloring, k-differential coloring, and equitable coloring. Equitable coloring of G is a vertex coloring of G that satisfies the condition that for each induced color class it has an equitable cardinality with difference 0 or 1. The minimum number of colors used for such coloring of G is called the equitable chromatic number of G, denoted by χe(G). In this study, we only concern with graphs that have a central vertex, which means a vertex that is adjacent to every other vertex, in particular on the star graph (Sn), lollipop graph (Ln), and friendship graph (fn). This research aims to formulate the equitable chromatic number of the star graph (Sn), lollipop graph (Ln), and friendship graph (fn). The first step taken in this research is to apply vertex coloring to Sn, Ln, and fn. After that, the color classes of the vertex set are obtained and its cardinality is determined. Next, analyze that the applied vertex coloring meets the definition of equitable coloring. Then, prove that the number of colors used is minimum. Thus, the chromatic number for each graph is obtained and proved. Based on this research, the equitable chromatic number of Sn is ⌈n/2⌉ + 1, the equitable chromatic number of Ln is n, and the equitable chromatic number of fn is 3, for n = 1 and n + 1, for n ≥ 2.

A proper edge coloring of a graph is an edge coloring such that no two adjacent edges are assigned the same color. The minimum number of colors needed to color edges of G is called the chromatic index χ/(G) of G. We introduce the new term Ce - V table for finding the inequality between and the sum of the terms domination number and the chromatic number of the induced sub graph. Here we introduced the applications of the edge coloring with the chromatic number in different dominations.