Long Alternating Cycles in Edge-Colored Complete Graphs

Some remarks on long monochromatic cycles in edge-colored complete graphs

Color degree and alternating cycles in edge-colored graphs

Given a graph G and an edge coloring C of G, an alternating cycle of G is such a cycle of G in which any adjacent edges have distinct colors. Let dc(v), named the color degree of a vertex v, be defined as the maximum number of edges incident with v, that have distinct colors. In this paper, some color degree conditions for the existence of alternating cycles of length 3 or 4 are obtained. We al...

Structure of Colored Complete Graphs Free of Proper Cycles

For a fixed integer m, we consider edge colorings of complete graphs which contain no properly edge colored cycle Cm as a subgraph. Within colorings free of these subgraphs, we establish a global structure by bounding the number of colors that can induce a spanning and connected subgraph. In the case of small cycles, namely C4, C5, and C6, we show that our bounds are sharp.

Degree-constrained spanning trees

S of the Ghent Graph Theory Workshop on Longest Paths and Longest Cycles Kathie Cameron Degree-constrained spanning trees 2 Jan Goedgebeur Finding minimal obstructions to graph coloring 3 Jochen Harant On longest cycles in essentially 4-connected planar graphs 3 Frantǐsek Kardoš Barnette was right: not only fullerene graphs are Hamiltonian 4 Gyula Y. Katona Complexity questions for minimally t-...

On Theorems Equivalent with Kotzig's Result on Graphs with Unique 1-Factors

We show that several known theorems on graphs and digraphs are equivalent. The list of equivalent theorems include Kotzig’s result on graphs with unique 1-factors, a lemma by Seymour and Giles, theorems on alternating cycles in edge-colored graphs, and a theorem on semicycles in digraphs. We consider computational problems related to the quoted results; all these problems ask whether a given (d...