Related papers: Packing six T-joins in plane graphs

We show that any planar graph $G=(V,E)$ has a 5-coloring such that one color class contains at most $|V|/6$ vertices. In other words, there exists a partition of $V$ into five independent sets $\{V_1, \cdots, V_5\}$ such that $|V_5| \leq…

For a finite graph $G$, we study the maximum $2$-edge colorable subgraph problem and a related ratio $\frac{\mu(G)}{\nu(G)}$, where $\nu(G)$ is the matching number of $G$, and $\mu(G)$ is the size of the largest matching in any pair…

Assume $G$ is a graph. We view $G$ as a symmetric digraph, in which each edge $uv$ of $G$ is replaced by a pair of opposite arcs $e=(u,v)$ and $e^{-1}=(v,u)$. Assume $S$ is an inverse closed subset of permutations of positive integers. We…

Let $P$ be a set of $n\geq 4$ points in general position in the plane. Consider all the closed straight line segments with both endpoints in $P$. Suppose that these segments are colored with the rule that disjoint segments receive different…

In his article [J. Comb. Theory Ser. B 16 (1974), 168-174], Tutte called two graphs $T$-equivalent (i.e., codichromatic) if they have the same Tutte polynomial and showed that graphs $G$ and $G'$ are $T$-equivalent if $G'$ is obtained from…

The famous K\H{o}nig-Egerv\'ary theorem is equivalent to the statement that the matching number equals the vertex cover number for every induced subgraph of some graph if and only if that graph is bipartite. Inspired by this result, we…

The {\em disjointness graph} $G=G({\cal S})$ of a set of segments ${\cal S}$ in $R^d$, $d\ge 2,$ is a graph whose vertex set is ${\cal S}$ and two vertices are connected by an edge if and only if the corresponding segments are disjoint. We…

A "dominating $K_t$-model" in a graph $G$ is a sequence $(T_1,\dots,T_t)$ of pairwise vertex-disjoint connected subgraphs of $G$, such that whenever $1\leq i<j\leq t$ every vertex in $T_j$ has a neighbour in $T_i$. Replacing "every vertex…

A $T$-decomposition of a graph $G$ is a set of edge-disjoint copies of $T$ in $G$ that cover the edge set of $G$. Graham and H\"aggkvist (1989) conjectured that any $2\ell$-regular graph $G$ admits a $T$-decomposition if $T$ is a tree with…

We show that for any colouring of the edges of the complete bipartite graph $K_{n,n}$ with 3 colours there are 5 disjoint monochromatic cycles which together cover all but $o(n)$ of the vertices. In the same situation, 18 disjoint…

A simpler proof of the four color theorem is presented. The proof was reached using a series of equivalent theorems. First the maximum number of edges of a planar graph is obatined as well as the minimum number of edges for a complete…

Let G be a combinatorial graph with vertices V and edges E. A proper coloring of G is an assignment of colors to the vertices such that no edge connects two vertices of the same color. These are the colorings considered in the famous Four…

We prove that if $H$ is a subgraph of a complete multipartite graph $G$, then $H$ contains a connected component $H'$ satisfying $|E(H')||E(G)|\geq |E(H)|^2$. We use this to prove that every three-coloring of the edges of a complete graph…

Thomassen famously proved that every planar graph is 5-choosable. We explore variants of this result, focusing on finding disjoint correspondence colorings, in the more general class of $K_5$-minor-free graphs. Correspondence colorings…

The Four Colour Theorem asserts that the vertices of every plane graph can be properly coloured with four colors. Fabrici and G\"oring conjectured the following stronger statement to also hold: the vertices of every plane graph can be…

It was conjectured by the third author in about 1973 that every $d$-regular planar graph (possibly with parallel edges) can be $d$-edge-coloured, provided that for every odd set $X$ of vertices, there are at least $d$ edges between $X$ and…

A path in an edge-colored graph is called a proper path if no two adjacent edges of the path receive the same color. For a connected graph $G$, the proper connection number $pc(G)$ of $G$ is defined as the minimum number of colors needed to…

As a natural extension of the Four Color Theorem, Haj\'{o}s conjectured that graphs containing no $K_5$-subdivision are 4-colorable. Any possible counterexample to this conjecture with minimum number of vertices is called a {\it Haj\'{o}s…

We show that any $2-$factor of a cubic graph can be extended to a maximum $3-$edge-colorable subgraph. We also show that the sum of sizes of maximum $2-$ and $3-$edge-colorable subgraphs of a cubic graph is at least twice of its number of…

Let $X_{2k}$ be a set of $2k$ labeled points in convex position in the plane. We consider geometric non-intersecting straight-line perfect matchings of $X_{2k}$. Two such matchings, $M$ and $M'$, are disjoint compatible if they do not have…