Suppose that T is a normal spanning tree (depth-first search tree) of a graph G. If e=xy and e′=uv are edges of G, satisfying x≺Tu≺Ty≺Tv, then they are called secant edges of G with respect to T. Suppose that G has no secant edges with respect to T. If T is a path, Ghazal and Al-Mniny proved that the chromatic number is at most 3. We conjecture that there is a positive constant γ such that, for any graph G that has no secant edges with respect to a normal spanning tree T, then χ(G)≤γ. We pose the problem of whether γ=4 suffices. We establish a positive answer in the case where T has at most one node.
An acyclic edge coloring of a graph G is a proper edge coloring such that G contains no bichromatic cycles. The acyclic chromatic index χa′(G) is the minimum number of colors required for an acyclic edge coloring. Fiamčik and Alon et al. independently conjectured that χa′(G)≤Δ+2 for every simple graph G with maximum de...
We introduce and begin the study of sequence b-colorings, a natural generalization of the classical notion of b-colorings introduced by Irving and Manlove in 1999. In a sequence b-coloring, each color class is required to contain a prescribed minimum number of color-dominating vertices (CDVs). We establish several fund...
Let G be a finite simple cubic graph, not necessarily connected, and let S_2(G) be obtained by subdividing every edge of G twice. Li (2025) developed general constructions for antimagic labelings of repeated subdivisions, but the cubic case G(3) = S_2(G) is not covered by those methods. Our first proof constructs an ed...
We combine two coloring aspects that work in opposite directions. One can 2-color the vertices of a cubic graph such that each monochromatic component is very small. One can also 2-color the vertices of a cubic graph such that each monochromatic component has degree at least 1. As an intended tool for solving a special...
A graph G is uniquely C_t-saturated if G contains no cycle of length t and, for every edge e of the complement, G+e contains exactly one cycle of length t; it is nontrivial if it has at least t vertices. Wenger and West proved that no nontrivial uniquely C_6- or C_7-saturated graphs exist, and conjectured the same for...
Fayzan Khan· 0 citations
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