Sharma and Panda recently proved that every bipartite graph with a perfect matching has property (P); that is, it admits a non-singular real symmetric matrix with support graph G for which every vertex is a P -vertex. In this paper, we extend their result from bipartite graphs to arbitrary graphs. To this end, we introduce the notion of a P - vertex covering and define the P -vertex covering number p(G) as the minimum number of non-singular matrices in S(G) needed so that every vertex of G is a P -vertex of at least one of them. Given a maximal matching of G, we partition the vertex set into the vertices saturated by the matching and the remaining vertices, which necessarily form an independent set. We then construct separate matrices covering these two classes of vertices. We use the Implicit Function Theorem as a perturbation tool to establish the desired result.
For a fixed graph F, the F-degree of a vertex v in a host graph H is the number of subgraphs of H isomorphic to F that contain v, and H is F-irregular if its F-degrees are pairwise distinct. We show that every finite connected graph F on at least three vertices admits a finite connected F-irregular host. For noncomplet...
We prove that the realization graph of every graphical degree sequence is maximally Hamiltonian: it is Hamilton-laceable when bipartite on more than one vertex, and Hamilton-connected otherwise. This answers Problem P59 of M\"utze's survey of combinatorial Gray codes, and the Hamiltonicity question recorded as open by...
Akbari, Elphick, Kumar, Pragada and Tang [Discrete Math. 349 (2026) 114953] conjectured that for every connected graph G, the line graph of G has at most one more positive than negative adjacency eigenvalue; equivalently, the signature of a connected line graph is at most 1. We refute the conjecture with two independen...
A graph is
perfect
if, for every induced subgraph, the chromatic number equals the size of its largest clique. In 1972, Lovász established a fundamental characterization of perfect graphs, showing that a graph is perfect if and only if, for every induced subgraph, the product of the size of the largest independen...
For a graph and a graph family , let denote the maximum number of copies of in an ‐free ‐vertex graph. Let . Bai, Tompkins, and Well conjectured that is attained if and each block of the graph is a . In this paper, we determine the exact value of and the extremal graphs for all . The novelty of our proof is to give a...
Xiaojun Zhao, Yuejian Peng· Journal of Graph Theory· 2 citations
The study of graph connectedness is a central topic in graph theory, with CF-connectedness being a specialized property of interest. A simple graph is CF-connected if it is connected and, in each of its optimal drawings, any two of its distinct vertices can be connected by a path consisting of uncrossed edges. This pap...
M. Staš, M. Švecová, Jana Fortes· Mathematics· 0 citations
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