Colton Magnant

Improved Upper Bounds for Gallai-Ramsey Numbers of Paths and Cycles

Given a graph G and a positive integer k, define the Gallai-Ramsey number to be the minimum number of vertices n such that any k-edge-coloring of Kn contains either a rainbow (all different colored) triangle or a monochromatic copy of G. In this work, we improve upon known upper bounds on the Gallai-Ramsey numbers for paths and cycles. All these upper bounds now have the best possible order of magnitude as functions of k.

Bounds on the Connected Forcing Number of a Graph

In this paper, we study (zero) forcing sets which induce connected subgraphs of a graph. The minimum cardinality of such a set is called the connected forcing number of the graph. We provide sharp upper and lower bounds on the connected forcing number in terms of the minimum degree, maximum degree, girth, and order of the graph.

Meander Graphs and Frobenius Seaweed Lie Algebras II

We provide a recursive classification of meander graphs, showing that each meander is identified by a unique sequence of fundamental graph theoretic moves. This sequence is called the meander’s signature and can be used to construct arbitrarily large sets of meanders, Frobenius or otherwise, of any size and configuration. In certain special cases, the signature is used to produce an explicit formula for the index of seaweed Lie subalgebra of sl(n) in terms of elementary functions.

Which tree has the smallest A B C index among trees with k leaves

Given a graph G , the atom-bond connectivity ( A B C ) index is defined to be A B C ( G ) = ? u ~ v d ( u ) + d ( v ) - 2 d ( u ) d ( v ) where u and v are vertices of G , d ( u ) denotes the degree of the vertex u , and u ~ v indicates that u and v are adjacent. Although it is known that among trees of a given order n , the star has maximum A B C index, we show that if k ? 18 , then the star of order k + 1 has minimum A B C index among trees with k leaves. If k ? 19 , then the balanced double star of order k + 2 has the smallest A B C index.

Multiply Chorded Cycles

A classical result of Hajnal and Szemeredi, when translated to a complementary form, states that with sufficient minimum degree, a graph will contain disjoint large cliques. We conjecture a generalization of this result from cliques to cycles with many chords and prove this conjecture in several cases.

General Bounds on Rainbow Domination Numbers

A k-rainbow dominating function of a graph G is a function f from the vertices V(G) to

Rainbow k-connection in Dense Graphs (Extended Abstract)

{2^{\{1, 2, \dots, k\}}}

On two conjectures about the proper connection number of graphs

2{1,2,⋯,k} such that, for all