Discussiones Mathematicae Graph Theory

Journal Article

The Existence of Path-Factor Covered Graphs

Dai, Guowei

2023 Discussiones Mathematicae Graph Theory

AbstractA spanning subgraph H of a graph G is called a P≥k-factor of G if every component of H is isomorphic to a path of order at least k, where k ≥ 2. A graph G is called a P≥k-factor covered graph if there is a P≥k-factor of G covering e for any e ∈ E(G). In this paper, we obtain two special...

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Journal Article

On Conditional Connectivity of the Cartesian Product of Cycles

Saraf, J.B.; Borse, Y.M.; Mundhe, Ganesh

2023 Discussiones Mathematicae Graph Theory

doi:10.7151/dmgt.2348

AbstractThe conditional h-vertex (h-edge) connectivity of a connected graph H of minimum degree k > h is the size of a smallest vertex (edge) set F of H such that H − F is a disconnected graph of minimum degree at least h. Let G be the Cartesian product of r ≥ 1 cycles, each of length at least...

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Journal Article

Unique Minimum Semipaired Dominating Sets in Trees

Haynes, Teresa W.; Henning, Michael A.

2023 Discussiones Mathematicae Graph Theory

doi:10.7151/dmgt.2349

AbstractLet G be a graph with vertex set V. A subset S ⊆ V is a semipaired dominating set of G if every vertex in V \ S is adjacent to a vertex in S and S can be partitioned into two element subsets such that the vertices in each subset are at most distance two apart. The semipaired domination...

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Journal Article

Burnside Chromatic Polynomials of Group-Invariant Graphs

White, Jacob A.

2023 Discussiones Mathematicae Graph Theory

doi:10.7151/dmgt.2385

AbstractWe introduce the Burnside chromatic polynomial of a graph that is invariant under a group action. This is a generalization of the Q-chromatic function Zaslavsky introduced for gain graphs. Given a group 𝕲 acting on a graph G and a 𝕲-set X, a proper X-coloring is a function with no...

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Journal Article

Equimatchable Bipartite Graphs

Büyükçolak, Yasemin; Gözüpek, Didem; Özkan, Sibel

2023 Discussiones Mathematicae Graph Theory

doi:10.7151/dmgt.2356

AbstractA graph is called equimatchable if all of its maximal matchings have the same size. Lesk et al. [Equi-matchable graphs, Graph Theory and Combinatorics (Academic Press, London, 1984) 239–254] has provided a characterization of equimatchable bipartite graphs. Motivated by the fact that this...

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Journal Article

Domination Parameters of the Unitary Cayley Graph of 𝕑/n𝕑

Burcroff, Amanda

2023 Discussiones Mathematicae Graph Theory

doi:10.7151/dmgt.2352

AbstractThe unitary Cayley graph of 𝕑/n𝕑, denoted Xn, is the graph with vertex set {0, . . ., n − 1} where vertices a and b are adjacent if and only if gcd(a − b, n) = 1. We answer a question of Defant and Iyer by constructing a family of infinitely many integers n such that γt(Xn) ≤ g(n) − 2,...

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Journal Article

Efficient (j, k)-Dominating Functions

Klostermeyer, William F.; MacGillivray, Gary; Semnani, Saeed Mohammadian; Piri, Farzaneh

2023 Discussiones Mathematicae Graph Theory

doi:10.7151/dmgt.2355

AbstractFor positive integers j and k, an efficient (j, k)-dominating function of a graph G = (V, E) is a function f : V → {0, 1, 2, . . ., j} such that the sum of function values in the closed neighbourhood of every vertex equals k. The relationship between the existence of efficient (j,...

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Journal Article

Elimination Properties for Minimal Dominating Sets of Graphs

Martí-Farré, Jaume; Mora, Mercè; Puertas, María Luz; Ruiz, José Luis

2023 Discussiones Mathematicae Graph Theory

doi:10.7151/dmgt.2354

AbstractA dominating set of a graph is a vertex subset such that every vertex not in the subset is adjacent to at least one in the subset. In this paper we study whenever there exists a new dominating set contained (respectively, containing) the subset obtained by removing a common vertex from...

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Journal Article

Improved Bounds for Some Facially Constrained Colorings

Štorgel, Kenny

2023 Discussiones Mathematicae Graph Theory

doi:10.7151/dmgt.2357

AbstractA facial-parity edge-coloring of a 2-edge-connected plane graph is a facially-proper edge-coloring in which every face is incident with zero or an odd number of edges of each color. A facial-parity vertex-coloring of a 2-connected plane graph is a proper vertex-coloring in which every...

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Journal Article

More Tales of Hoffman: Bounds for the Vector Chromatic Number of a Graph

Wocjan, Pawel; Elphick, Clive; Anekstein, David

2023 Discussiones Mathematicae Graph Theory

doi:10.7151/dmgt.2358

AbstractLet χ(G) denote the chromatic number of a graph and χv(G) denote the vector chromatic number. For all graphs χv(G) ≤ χ(G) and for some graphs χv(G) ≪ χ(G). Galtman proved that Hoffman’s well-known lower bound for χ(G) is in fact a lower bound for χv(G). We prove that two more spectral...

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