Remarks on Graphons

The notion of the graphon (a symmetric measurable fuzzy set of $[0, 1]^2$) was introduced by L. Lov\'asz and B. Szegedy in 2006 to describe limit objects of convergent sequences of dense graphs. In their investigation the integral $$t(F,W)=\int _{[0, 1]^k}\prod _{ij\in E(F)}W(x_i,x_j)dx_1dx_2\cdots dx_k$$ plays an important role in which $W$ is a graphon and $E(F)$ denotes the set of all edges of a $k$-labelled simple graph $F$. In our present paper we show that the set of all fuzzy sets of $[0, 1]^2$ is a right regular band with respect to the operation $\circ$ defined by $$(f\circ g)(s,t)=\vee _{(x,y)\in [0, 1]^2}(f(x,y)\wedge g(s,t));\quad (s, t)\in [0, 1]^2,$$ and the set of all graphons is a left ideal of this band. We prove that, if $W$ is an arbitrary graphon and $f$ is a fuzzy set of $[0, 1]^2$, then $$|t(F; W)-t(F; f\circ W)|\leq |E(F)|(\sup(W)-\sup(f))\Delta (\{W> \sup(f)\} )$$ for arbitrary finite simple graphs $F$, where $\Delta (\{W> \sup(f)\})$ denotes the area of the set $\{W>\sup(f)\}$ of all $(x, y)\in [0, 1]^2$ satisfying $W(x,y)>\sup(f)$.Comment: 11 page

Degree and neighborhood conditions for hamiltonicity of claw-free graphs

For a graph H , let σ t ( H ) = min { Σ i = 1 t d H ( v i ) | { v 1 , v 2 , … , v t } is an independent set in H } and let U t ( H ) = min { | ⋃ i = 1 t N H ( v i ) | | { v 1 , v 2 , ⋯ , v t } is an independent set in H } . We show that for a given number ϵ and given integers p ≥ t \u3e 0 , k ∈ { 2 , 3 } and N = N ( p , ϵ ) , if H is a k -connected claw-free graph of order n \u3e N with δ ( H ) ≥ 3 and its Ryjác̆ek’s closure c l ( H ) = L ( G ) , and if d t ( H ) ≥ t ( n + ϵ ) ∕ p where d t ( H ) ∈ { σ t ( H ) , U t ( H ) } , then either H is Hamiltonian or G , the preimage of L ( G ) , can be contracted to a k -edge-connected K 3 -free graph of order at most max { 4 p − 5 , 2 p + 1 } and without spanning closed trails. As applications, we prove the following for such graphs H of order n with n sufficiently large: (i) If k = 2 , δ ( H ) ≥ 3 , and for a given t ( 1 ≤ t ≤ 4 ), then either H is Hamiltonian or c l ( H ) = L ( G ) where G is a graph obtained from K 2 , 3 by replacing each of the degree 2 vertices by a K 1 , s ( s ≥ 1 ). When t = 4 and d t ( H ) = σ 4 ( H ) , this proves a conjecture in Frydrych (2001). (ii) If k = 3 , δ ( H ) ≥ 24 , and for a given t ( 1 ≤ t ≤ 10 ) d t ( H ) \u3e t ( n + 5 ) 10 , then H is Hamiltonian. These bounds on d t ( H ) in (i) and (ii) are sharp. It unifies and improves several prior results on conditions involved σ t and U t for the hamiltonicity of claw-free graphs. Since the number of graphs of orders at most max { 4 p − 5 , 2 p + 1 } are fixed for given p , improvements to (i) or (ii) by increasing the value of p are possible with the help of a computer

Minimum Degrees of Minimal Ramsey Graphs for Almost-Cliques

For graphs $F$ and $H$, we say $F$ is Ramsey for $H$ if every $2$-coloring of the edges of $F$ contains a monochromatic copy of $H$. The graph $F$ is Ramsey $H$-minimal if $F$ is Ramsey for $H$ and there is no proper subgraph $F'$ of $F$ so that $F'$ is Ramsey for $H$. Burr, Erdos, and Lovasz defined $s(H)$ to be the minimum degree of $F$ over all Ramsey $H$-minimal graphs $F$. Define $H_{t,d}$ to be a graph on $t+1$ vertices consisting of a complete graph on $t$ vertices and one additional vertex of degree $d$. We show that $s(H_{t,d})=d^2$ for all values $1<d\le t$; it was previously known that $s(H_{t,1})=t-1$, so it is surprising that $s(H_{t,2})=4$ is much smaller. We also make some further progress on some sparser graphs. Fox and Lin observed that $s(H)\ge 2\delta(H)-1$ for all graphs $H$, where $\delta(H)$ is the minimum degree of $H$; Szabo, Zumstein, and Zurcher investigated which graphs have this property and conjectured that all bipartite graphs $H$ without isolated vertices satisfy $s(H)=2\delta(H)-1$. Fox, Grinshpun, Liebenau, Person, and Szabo further conjectured that all triangle-free graphs without isolated vertices satisfy this property. We show that $d$-regular $3$-connected triangle-free graphs $H$, with one extra technical constraint, satisfy $s(H) = 2\delta(H)-1$; the extra constraint is that $H$ has a vertex $v$ so that if one removes $v$ and its neighborhood from $H$, the remainder is connected.Comment: 10 pages; 3 figure

Intersection Numbers of Geodesic Arcs

For a compact surface $S$ with constant negative curvature $-\kappa$ (for some $\kappa>0$) and genus $g\geq2$, we show that the tails of the distribution of $i(\alpha,\beta)/l(\alpha)l(\beta)$ (where $i(\alpha,\beta)$ is the intersection number of the closed geodesics and $l(\cdot)$ denotes the geometric length) are estimated by a decreasing exponential function. As a consequence, we find the asymptotic normalized average of the intersection numbers of pairs of closed geodesics on $S$. In addition, we prove that the size of the sets of geodesics whose $T$-self-intersection number is not close to $\kappa T^2/(2\pi^2(g-1))$ is also estimated by a decreasing exponential function. And, as a corollary of the latter, we obtain a result of S. Lalley which states that most of the closed geodesics $\alpha$ on $S$ with $l(\alpha)\leq T$ have roughly $\kappa l(\alpha)^2/(2\pi^2(g-1))$ self-intersections, when $T$ is large