The Maximum Number of Triangles in Graphs Without Cycles of Length 0mod5

Cycles in Unions of Transitive Tournaments

A Note on Lovász Characterization of Perfect Graphs

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 independent set and the size of the largest clique is at least the number of vertices. His proof relied on the technique of vertex replication. In this paper, we present an alternative proof of Lovász's result that avoids vertex replication. As vertex replication does not in general preserve ‐perfection, the argument developed here applies to the study of ‐perfect graphs, a class introduced by Ravindra in 2011.

The Maximum Number of Shortest Paths in Graphs

Benjamini and Tzalik obtained an upper bound on the number of shortest paths between two vertices at distance $t$ in a multigraph of maximum degree at most $\Delta$, and proposed a conjecture on the sharp bound. In this paper, we develop a probabilistic counting argument based on probability distributions induced by random walks from the two endpoints. This approach yields a sharp bound for multigraphs and confirms their conjecture. We further determine the exact maximum for simple graphs and thus answer another question of Benjamini and Tzalik. We also investigate the equality cases, describing the structure of the subgraph formed by shortest paths between $x$ and $y$ and giving tight examples.

Total Neighborhood Prime Labeling of Some Graphs

Packing and Covering Cycles Through Prescribed Vertices

Let $G$ be a finite simple graph and let $S\subseteq V(G)$. We prove that the minimum number of vertices meeting every cycle that intersects $S$ is at most the maximum number of vertices of $S$ covered by a collection of vertex-disjoint cycles. This answers a question posed by Bowler, Ghorbani, Gut, Jacobs, and Reich [\emph{SIAM Journal on Discrete Mathematics} \textbf{40} (2026), 988--999]. An incidence-based reduction to their bidirected packing--covering theorem preserves the packing value and projects transversals without increasing their cardinality.

On r-dynamic coloring of graphs in subclasses of planar and circulant graphs

An r-dynamic coloring of a graph G is a proper vertex coloring in which each vertex sees at least min{r, d(v)} distinct colors in its neighborhood. The minimum number of colors in such a coloring is the r-dynamic chromatic number χdr(G). We determine exact values and upper bounds of χdr for several graph classes, including triangular grids, planar 3-trees for r ≤ 4, and planar Eulerian triangulations for r ≤ 3 (with a partial result for r = 4), confirming the conjecture of [Song et al. 2014] for these subclasses. We also establish exact values for the 2-dynamic chromatic number of a subclass of circulant graphs, confirming a conjecture of [Montgomery 2001] for this regular family.