We study the problem of determining whether a given graph G = (V,E) admits a matching M whose removal destroys all odd cycles of G (or equivalently whether G−M is bipartite). This problem is also equivalent to determine whether G admits a (1,1)-coloring, which is a 2-coloring of V (G) in which each color class induces a graph of maximum degree at most 1. We show that such a decision problem is ...

‎for any integer $kgeq 1$‎, ‎a set $s$ of vertices in a graph $g=(v,e)$ is a $k$-‎tuple total dominating set of $g$ if any vertex‎ ‎of $g$ is adjacent to at least $k$ vertices in $s$‎, ‎and any vertex‎ ‎of $v-s$ is adjacent to at least $k$ vertices in $v-s$‎. ‎the minimum number of vertices of such a set‎ ‎in $g$ we call the $k$-tuple total restrained domination number of $g$‎. ‎the maximum num...

Systems of polynomial equations are commonly used to model combinatorial problems such as independent set, graph coloring, Hamiltonian path, and others. We formulate the dominating set problem as a system of polynomial equations in two different ways: first, as a single, high-degree polynomial, and second as a collection of polynomials based on the complements of domination-critical graphs. We ...

We consider a multiagent extension of single-agent graph coloring. Multiple agents hold disjoint autonomous subgraphs of a global graph, and every color used by the agents in coloring the graph has associated cost. In this multiagent graph coloring scenario, we seek a minimum legal coloring of the global graph’s vertices, such that the coloring is also Pareto efficient, socially fair, and indiv...

Many classes of graphs where the vertex coloring problem is polynomially solvable are known, the most prominent being the class of perfect graphs. However, the list-coloring problem is NP-complete for many subclasses of perfect graphs. In this work we explore the complexity boundary between vertex coloring and list-coloring on such subclasses of perfect graphs, where the former admits polynomia...

A hypergraph is b-simple if no two distinct edges share more than b vertices. Let m(r, t, g) denote the minimum number of edges in an r-uniform non-t-colorable hypergraph of girth at least g. Erdős and Lovász proved that m(r, t, 3) ≥ t 2(r−2) 16r(r − 1)2 and m(r, t, g) ≤ 4 · 20g−1r3g−5t(g−1)(r+1). A result of Szabó improves the lower bound by a factor of r2− for sufficiently large r. We improve...