Triangles in graphs without bipartite suspensions

Given graphs $T$ and $H$, the generalized Tur\'an number ex$(n,T,H)$ is the maximum number of copies of $T$ in an $n$-vertex graph with no copies of $H$. Alon and Shikhelman, using a result of Erd\H os, determined the asymptotics of ex$(n,K_3,H)$ when the chromatic number of $H$ is greater than 3 and proved several results when $H$ is bipartite. We consider this problem when $H$ has chromatic number 3. Even this special case for the following relatively simple 3-chromatic graphs appears to be challenging. The suspension $\widehat H$ of a graph $H$ is the graph obtained from $H$ by adding a new vertex adjacent to all vertices of $H$. We give new upper and lower bounds on ex$(n,K_3,\widehat{H})$ when $H$ is a path, even cycle, or complete bipartite graph. One of the main tools we use is the triangle removal lemma, but it is unclear if much stronger statements can be proved without using the removal lemma.Comment: New result about path with 5 edges adde

Nombre chromatique fractionnaire, degré maximum et maille

We prove new lower bounds on the independence ratio of graphs of maximum degree ∆ ∈ {3,4,5} and girth g ∈ {6,…,12}, notably 1/3 when (∆,g)=(4,10) and 2/7 when (∆,g)=(5,8). We establish a general upper bound on the fractional chromatic number of triangle-free graphs, which implies that deduced from the fractional version of Reed's bound for triangle-free graphs and improves it as soon as ∆ ≥ 17, matching the best asymptotic upper bound known for off-diagonal Ramsey numbers. In particular, the fractional chromatic number of a triangle-free graph of maximum degree ∆ is less than 9.916 if ∆=17, less than 22.17 if ∆=50 and less than 249.06 if ∆=1000. Focusing on smaller values of ∆, we also demonstrate that every graph of girth at least 7 and maximum degree ∆ has fractional chromatic number at most min (2∆ + 2^{k-3}+k)/k pour k ∈ ℕ. In particular, the fractional chromatic number of a graph of girth 7 and maximum degree ∆ is at most (2∆+9)/5 when ∆ ∈ [3,8], at most (∆+7)/3 when ∆ ∈  [8,20], at most (2∆+23)/7 when ∆ ∈ [20,48], and at most ∆/4+5 when ∆ ∈ [48,112]

On the ultimate normalized chromatic difference sequence of a graph

AbstractFor graphs G and H, the Cartesian product G × H is defined as follows: the vertex set is V(G) × V(H), and two vertices (g,h) and (g′,h′) are adjacent in G × H if either g = g′ and hh′ ϵ E(H) or h = h′ and gg′ ϵ E(G). Let Gk denote the Cartesian product of k copies of G. The chromatic difference sequence cds(G) is defined by cds(G) = (a1, a2 − a1,…, at − at − 1,…) where at denotes the maximum number of vertices of t-colorable subgraph of G. The normalized chromatic difference sequence ncds(G) is defined by ncds(G) = cds(G)/V(G). This paper studies the ultimate normalized chromatic difference sequence of a graph NCDS(G) which is equal to the limit of ncds(Gk) as k goes to infinity. We study NCDS(G) under the context of other graph theoretical properties: star chromatic number, hom-regularity, and graph homomorphism. We have provided new upper and lower bounds for NCDS(G). We have also proved, among others, that if there is a homomorphism from a graph G to a graph H, then NCDS(G) dominates NCDS(H)

On (d,1)-total numbers of graphs

AbstractA (d,1)-total labelling of a graph G assigns integers to the vertices and edges of G such that adjacent vertices receive distinct labels, adjacent edges receive distinct labels, and a vertex and its incident edges receive labels that differ in absolute value by at least d. The span of a (d,1)-total labelling is the maximum difference between two labels. The (d,1)-total number, denoted λdT(G), is defined to be the least span among all (d,1)-total labellings of G. We prove new upper bounds for λdT(G), compute some λdT(Km,n) for complete bipartite graphs Km,n, and completely determine all λdT(Km,n) for d=1,2,3. We also propose a conjecture on an upper bound for λdT(G) in terms of the chromatic number and the chromatic index of G