Almost eulerian compatible spanning circuits in edge-colored graphs

Eulerian Circuits with No Monochromatic Transitions in Edge-colored Digraphs

Let G be an eulerian digraph with a fixed edge coloring (not necessarily a proper edge coloring). A compatible circuit of G is an eulerian circuit such that every two consecutive edges in the circuit have different colors. We characterize the existence of compatible circuits for directed graphs avoiding certain vertices of outdegree three. Our result is analogous to a result of Kotzig for compa...

Rainbow spanning subgraphs of edge-colored complete graphs

Consider edge-colorings of the complete graph Kn. Let r(n, t) be the maximum number of colors in such a coloring that does not have t edge-disjoint rainbow spanning trees. Let s(n, t) be the maximum number of colors in such a coloring having no rainbow spanning subgraph with diameter at most t. We prove r(n, t) = (n−2 2 )

Multi-colored Spanning Graphs

We study a problem proposed by Hurtado et al. [10] motivated by sparse set visualization. Given n points in the plane, each labeled with one or more primary colors, a colored spanning graph (CSG) is a graph such that for each primary color, the vertices of that color induce a connected subgraph. The Min-CSG problem asks for the minimum sum of edge lengths in a colored spanning graph. We show th...

Almost Spanning Subgraphs of Random Graphs After Adversarial Edge Removal

Let ∆ ≥ 2 be a fixed integer. We show that the random graph Gn,p with p ≥ c(logn/n)1/∆is robust with respect to the containment of almost spanning bipartite graphs H with maximum degree ∆and sublinear bandwidth in the following sense. If an adversary deletes arbitrary edges in Gn,p such thateach vertex loses less than half of its neighbours, then asymptotically almost surely the res...

Sufficient conditions for the existence of spanning colored trees in edge-colored graphs