Об индексе палитры треугольника Серпинского и графа Серпинского

The palette of a vertex v of a graph G in a proper edge coloring is the set of colors assigned to the edges which are incident to v. The palette index of G is the minimum number of palettes occurring among all proper edge colorings of G. In this paper, we consider the palette index of Sierpinski graphs S” and Sierpinski triangle graphs S” . In particular, we determine the exact value of the palette index of Sierpinski triangle graphs. We also determine the palette index of Sierpinski graphs S” where p is even, p = 3, or n = 2 and p = 41 + 3

Covering codes in Sierpinski graphs

Graphs and AlgorithmsInternational audienceFor a graph G and integers a and b, an (a, b)-code of G is a set C of vertices such that any vertex from C has exactly a neighbors in C and any vertex not in C has exactly b neighbors in C. In this paper we classify integers a and b for which there exist (a, b)-codes in Sierpinski graphs

Selected Problems in Graph Coloring

The Borodin–Kostochka Conjecture states that for a graph G, if ∆(G) ≥ 9 and ω(G) ≤ ∆(G) − 1, then χ(G) ≤ ∆(G) − 1. We prove the Borodin–Kostochka Conjecture for (P5, gem)-free graphs, i.e., graphs with no induced P5 and no induced K1 ∨P4. ForagraphGandt,k∈Z+ at-tonek-coloringofGisafunctionf:V(G)→ [k] such that |f(v)∩f(w)| \u3c d(v,w) for all distinct v,w ∈ V(G). The t-tone t chromatic number of G, denoted τt(G), is the minimum k such that G is t-tone k- colorable. For small values of t, we prove sharp or nearly sharp upper bounds on the t-tone chromatic number of various classes of sparse graphs. In particular, we determine τ2(G) exactly when mad(G) \u3c 12/5 and also determine τ2(G), up to a small additive constant, when G is outerplanar. Finally, we determine τt(Cn) exactly when t ∈ {3, 4, 5}

Graph Theory

Highlights of this workshop on structural graph theory included new developments on graph and matroid minors, continuous structures arising as limits of finite graphs, and new approaches to higher graph connectivity via tree structures

On Chromatic Polynomial and Ordinomial

Not included

Feedback vertex number of Sierpi\'{n}ski-type graphs

The feedback vertex number $\tau(G)$ of a graph $G$ is the minimum number of vertices that can be deleted from $G$ such that the resultant graph does not contain a cycle. We show that $\tau(S_p^n)=p^{n-1}(p-2)$ for the Sierpi\'{n}ski graph $S_p^n$ with $p\geq 2$ and $n\geq 1$. The generalized Sierpi\'{n}ski triangle graph $\hat{S_p^n}$ is obtained by contracting all non-clique edges from the Sierpi\'{n}ski graph $S_p^{n+1}$. We prove that $\tau(\hat{S}_3^n)=\frac {3^n+1} 2=\frac{|V(\hat{S}_3^n)|} 3$, and give an upper bound for $\tau(\hat{S}_p^n)$ for the case when $p\geq 4$