Cover numbers by certain graph families
Márton Marits
TL;DR
The paper introduces the cover number c_P(G) as the minimum count of graphs from a class P whose edge-sets cover E(G). It proves an exact formula for covers by chi-bounded classes $\{χ \le f(ω)\}$, namely $c_{\{χ \le f(ω)\}}(G) = \left\lceil \frac{\log χ(G)}{\log f(ω(G))} \right\rceil$, and relates it to the classical bipartite case via $c_{\mathbf{BIP}}(G) = \left\lceil \log_2 χ(G) \right\rceil$. The work constructs a chain of inequalities among cover numbers for perfect, generalized split, co-unipolar, and bipartite graphs and shows that gaps between successive classes can be arbitrarily large. It also demonstrates that some intermediate cover numbers cannot be expressed solely in terms of $χ$ and $ω$ by providing explicit constructions (e.g., hypercubes, Zykov-type graphs) that separate the classes. Open problems include determining exact values for several classes and characterizing when a function f can serve as a cover-number function for some graph class.
Abstract
We define the cover number of a graph $G$ by a graph class $\mathcal P$ as the minimum number of graphs of class $\mathcal P$ required to cover the edge set of $G$. Taking inspiration from a paper by Harary, Hsu and Miller, we find an exact formula for the cover number by the graph classes $\{ G \mid χ(G) \leq f(ω(G))\}$ for an arbitrary non-decreasing function $f$. After this, we establish a chain of inequalities with five cover numbers, the one by the class $\{ G \mid χ(G) = ω(G)\}$, by the class of perfect graphs, generalized split graphs, co-unipolar graphs and finally by bipartite graphs. We prove that at each inequality, the difference between the two sides can grow arbitrarily large. We also prove that the cover number by unipolar graphs cannot be expressed in terms of the chromatic or the clique number.