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Abstract: Graph burning is a round-based game or process that discretely models the spread of influence throughout a network. We introduce a generalization of graph burning which applies to hypergraphs, as well as a variant called "lazy" hypergraph burning. Interestingly, lazily burning a graph is trivial, while lazily burning a hypergraph can be quite complicated. Moreover, the lazy burning model is a useful tool for analyzing the round-based model. One of our key results is that arbitrary hypergraphs do not satisfy a bound analogous to the one in the Burning Number Conjecture for graphs. We also obtain bounds on the burning number and lazy burning number of a hypergraph in terms of its parameters, and present several open problems in the field of (lazy) hypergraph burning.
Keywords: Combinatorial games on graphs, pursuit-evasion, graph searching, graph burning, hypergraph theory
Published in RUP: 18.08.2026; Views: 300; Downloads: 5
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Abstract: A ({3, m}; g)-semicubic graph is a graph where the degree of each vertex is either 3 or m and the girth of the graph is g; if m = 3 we have a cubic graph. In this paper, we construct families of semicubic graphs of even girth and small order using two different techniques. The first technique generalizes a previous construction, which glues cubic cages of girth g together at remote vertices (vertices at distance at least g/2). The second technique, the main content of this paper, produces bipartite semicubic ({3, m}; g)-graphs of even girth g using voltage graphs over ℤm. For girth g = 4t + 2, t ≥ 1, the constructed graphs have two vertices of degree m. For girth g = 4t, t ≥ 2, the construction produces graphs with exactly three vertices of degree m (of course, the remaining vertices are of degree 3 in both cases). In particular, we describe infinite families of ({3, m}; g)−semicubic graphs for g = {6, 8, 10, 12} for infinitely many values of m. The cases g = {6, 8} include the unique 6-cage and the unique 8-cage when m = 3. The families obtained in this paper for girth g = {10, 12} include examples of orders that match the best-known bounds for ({3, m}; g)−semicubic graphs until this moment.
Keywords: Graph, semicubic graph, girth, voltage graph
Published in RUP: 17.08.2026; Views: 189; Downloads: 4
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Abstract: Since Reed conjectured in 1996 that the domination number of a connected cubic graph of order n is at most ⌈1/3n⌉, the domination number of cubic graphs has been extensively studied. It is now known that the conjecture is false in general, but Henning and Dorbec showed that it holds for graphs with girth at least 9. Zhu and Wu stated an analogous conjecture for 2-connected cubic planar graphs. In this paper, we present a new upper bound for the domination number of subcubic planar graphs: if G is a subcubic planar graph with girth at least 8, then γ(G) < n₀ + 3/4 n₁ + 11/20 n₂ + 7/20 n₃, where n_i denotes the number of vertices in G of degree i, for i ∈ {0, 1, 2, 3}. We also prove that if G is a subcubic planar graph with girth at least 9, then γ(G) < n₀ + 13/17 n₁ + 9/17 n₂ + 6/17 n₃.
Keywords: domination, subcubic planar graph, upper bound
Published in RUP: 11.08.2026; Views: 207; Downloads: 3
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Abstract: A nut graph is a nontrivial simple graph whose adjacency matrix has a one-dimensionalnull space such that its nonzero vectors contain no zero elements. For circulant graphs, itis known that they are nut if and only if their nullity is one. This fact was recently usedby the author in order to show that there exists ad-regular circulant nut graph of order n if and only if 4|d,2|n, d >0, alongside n ≥d+ 4 if d≡84, and n ≥d+ 6 if 8|d, and (n, d)̸= (16,8). Here, we deal with the quartic circulant graphs and disclose several results regarding their null spaces. First of all, we derive an explicit formula for computing the nullity of any quartic circulant graph. Furthermore, we provide the full nut graph characterization among these graphs. Finally, we determine all such graphs that attain the minimum or maximum nullity with respect to a given order. We also give the extremal null spaces of these graphs.
Keywords: circulant graph, quartic graph, null space, nut graph, singular graph, adjacency matrix
Published in RUP: 27.07.2026; Views: 258; Downloads: 5
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Abstract: Chemical reactions can be understood as transformations of multigraphs (molecules)that preserve vertex labels (atoms) and degrees (sums of bonding and non-bonding electrons), thereby implying the atom-atom map of a reaction. The corresponding reaction mechanism is often described by an electron pushing diagram that explains the transformation by consecutive local relocations of individual edges (electron pairs). Here, we showthat every degree-preserving map between multigraphs, and thus every atom-atom map, can be generated by cyclic electron pushing. Moreover, it is always possible to decompose such an explanation into electron pushing diagrams involving only four electron pairs. This inturn implies that every reaction can be decomposed into a sequence of elementary reactions that involve at most two educt molecules and two product molecules. Hence, the requirement of a mechanistic explanation in terms of electron pushing and imaginary transition states involving only a small number of bonds does not impose a combinatorial constraint on the feasibility of hypothetical chemical reactions.
Keywords: chemical reaction networks, graph transformations, reaction mechanisms
Published in RUP: 27.07.2026; Views: 294; Downloads: 9
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