相关论文: Francis Guthrie's approach to The Four Color Probl…

A formal proof has not been found for the four color theorem since 1852 when Francis Guthrie first conjectured the four color theorem. Why? A bad idea, we think, directed people to a rough road. Using a similar method to that for the formal…

The famous four color theorem states that for all planar graphs, every vertex can be assigned one of 4 colors such that no two adjacent vertices receive the same color. Since Francis Guthrie first conjectured it in 1852, it is until 1976…

The four-color theorem states that no more than four colors are required to color all nodes in planar graphs such that no two adjacent nodes are of the same color. The theorem was first propounded by Francis Guthrie in 1852. Since then,…

A proper $k$-coloring of $G$ is called an odd coloring of $G$ if for every vertex $v$, there is a color that appears at an odd number of neighbors of $v$. This concept was introduced recently by Petru\v{s}evski and \v{S}krekovski, and they…

Acceptable but due to extensive usage of a computer rather unpleasant proof of the famous four color map problem of Francis Guthrie were settled eventually by W. Appel and K. Haken in 1976. Using the same method but shortening the proof…

A proper vertex coloring $\varphi$ of graph $G$ is said to be odd if for each non-isolated vertex $x\in V(G)$ there exists a color $c$ such that $\varphi^{-1}(c)\cap N(x)$ is odd-sized. The minimum number of colors in any odd coloring of…

An {\em odd subgraph} of a graph is a subgraph in which every vertex has odd degree. A graph $G$ is said to be {\em odd $k$-edge-colorable} if there exists an edge-coloring $E(G) \rightarrow \{1,2, \ldots, k\}$ such that each non-empty…

For a graph $F$, let ${\rm EX}(n,F)$ be the set of $F$-free graphs of order $n$ with the maximum number of edges. The graph $F$ is called vertex-critical, if the deletion of its some vertex induces a graph with smaller chromatic number. For…

Odd coloring is a variant of proper coloring and has received wide attention. We study the list-coloring version of this notion in this paper. We prove that if $G$ is a graph embeddable in the torus or the Klein bottle with no cycle of…

The Four color problem is closely related to other branches of mathematics and practical applications. More than 20 of its reformulations are known, which connect this problem with problems of algebra, statistical mechanics and planning.…

An {\it odd $c$-coloring} of a graph is a proper $c$-coloring such that each non-isolated vertex has a color appearing an odd number of times on its neighborhood. This concept was introduced very recently by Petru\v sevski and \v Skrekovski…

An odd graph is a finite graph all of whose vertices have odd degrees. Given graph $G$ is decomposable into $k$ odd subgraphs if its edge set can be partitioned into $k$ subsets each of which induces an odd subgraph of $G$. The minimum…

In a simple graph $G$, we prove that the \textit{Hadwiger number}, $h(G)$, of the given graph $G$ always upper bounds the \textit{chromatic number}, $\chi(G)$, of the given graph $G$, that is, $\chi(G) \leq h(G)$. This simply stated problem…

A graph is called odd (respectively, even) if every vertex has odd (respectively, even) degree. Gallai proved that every graph can be partitioned into two even induced subgraphs, or into an odd and an even induced subgraph. We refer to a…

A proper coloring of a graph is odd if every non-isolated vertex has some color that appears an odd number of times on its neighborhood. This notion was recently introduced by Petru\v{s}evski and \v{S}krekovski, who proved that every planar…

A proper vertex-colouring of a simple graph $G$ is said to be odd if, for every non-isolated vertex $v$ of $G$, some colour appears an odd number of times in the neighbourhood of $v$. We show that if $G$ embeds in the torus, then it admits…

For a graph (undirected, directed, or mixed), a cycle-factor is a collection of vertex-disjoint cycles covering the entire vertex set. Cycle-factors subject to parity constraints arise naturally in the study of structural graph theory and…

A strong edge coloring of a graph is a proper edge coloring in which every color class is an induced matching. The strong chromatic index $\chi_s'(G)$ of a graph $G$ is the minimum number of colors in a strong edge coloring of $G$. Let…

Petru\v{s}evski and \v{S}krekovski \cite{odd9} recently introduced the notion of an odd colouring of a graph: a proper vertex colouring of a graph $G$ is said to be \emph{odd} if for each non-isolated vertex $x \in V(G)$ there exists a…

Odd coloring is a proper coloring with an additional restriction that every non-isolated vertex has some color that appears an odd number of times in its neighborhood. The minimum number of colors $k$ that can ensure an odd coloring of a…