Search Results

In graph theory, strong edge coloring is a proper edge coloring in which every two edges with adjacent endpoints must have different colors. A 1-subdivision of a graph results from inserting 1 new ver …

Who knows the name of the following coloring of graphs, a proper vertex coloring so that for every vertex its every two neighbors receive different colors?

A $b$-fold coloring of a graph G is an assignment of sets of size $b$ to vertices of a graph such that adjacent vertices receive disjoint sets. An $a:b$-coloring is a $b$-fold coloring out of $a$ avai …

Yes! Let $uv$ be an edge colored by the last color, say $\Delta+1$. If $uv$ is incidence with all colors, then it is the required edge. So the only case is that every edge colored with $\Delta+1$ is n …

I do not think there is a standard name for this, but I may prefer to call it $p$-random chromatic number....

A proper vertex coloring of a graph $G$ is acyclic if there is no bicolored cycle. A graph is 2-degenerate if its every subgraph has a vertex of degree at most 2. I think every 2-degenerate graph has …

Edge coloring of a graph is an assignment of “colors” to the edges of the graph so that no two adjacent edges have the same color with an optimal number of colors. Two edges are said to be adjacent if …

A 2-planar graph is a graph that can be drawn in the plane so that each edge is crossed at most twice. It is known that every 2-planar graph satisfies that $|E(G)|\le 5(|V(G)|-2)$. This implies that e …

A proper edge coloring is a coloring of the edges of a graph so that adjacent edges receive distinct colors. Vizing's theorem states that every simple graph $G$ has a proper edge coloring using at mos …

By a classical theorem of Nash-Williams, the edges of every connected $n$-vertex planar graph can be covered by three trees $T_1,T_2$ and $T_3$. Does anyone know of any results from an article or a te …

Given a multigraph such that there are 0 or 2 edges connecting every two vertices, we are to color the edges of this graph so that adjacent edges receive distinct colors. It is known that we need at l …

A toroidal graph is a graph that can be embedded on a torus. In other words, the graph's vertices can be placed on a torus such that no edges cross. A minor of graph G is a graph obtained from G by me …

A well-known conjecture of Berge and Fulkerson says that every bridgeless cubic graph has a collection of six perfect matchings that together cover every edge exactly twice. Is this still open for bri …