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Register a new userAn upper bound on the extremal version of Hajnals triangle-free game
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A game starts with the empty graph on $n$ vertices, and two player alternate adding edges to the graph. Only moves which do not create a triangle are valid. The game ends when a maximal triangle-free graph is reached. The goal of one player is to end the game as soon as possible, while the other player is trying to prolong the game. With optimal play, the length of the game (number of edges played) is called the $K_3$ game saturation number. In this paper we prove an upper bound for this number.
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We show every triangle-free $4$-critical graph $G$ satisfies $e(G) geq frac{5v(G)+2}{3}$.
The Wiener index of a connected graph is the summation of all distances between unordered pairs of vertices of the graph. In this paper, we give an upper bound on the Wiener index of a $k$-connected graph $G$ of order $n$ for integers $n-1>k ge 1$: [W(G) le frac{1}{4} n lfloor frac{n+k-2}{k} rfloor (2n+k-2-klfloor frac{n+k-2}{k} rfloor).] Moreover, we show that this upper bound is sharp when $k ge 2$ is even, and can be obtained by the Wiener index of Harary graph $H_{k,n}$.
An orientation of a graph is semi-transitive if it is acyclic, and for any directed path $v_0rightarrow v_1rightarrow cdotsrightarrow v_k$ either there is no arc between $v_0$ and $v_k$, or $v_irightarrow v_j$ is an arc for all $0leq i<jleq k$. An undirected graph is semi-transitive if it admits a semi-transitive orientation. Semi-transitive graphs generalize several important classes of graphs and they are precisely the class of word-representable graphs studied extensively in the literature. Determining if a triangle-free graph is semi-transitive is an NP-hard problem. The existence of non-semi-transitive triangle-free graphs was established via ErdH{o}s theorem by Halld{o}rsson and the authors in 2011. However, no explicit examples of such graphs were known until recent work of the first author and Saito who have shown computationally that a certain subgraph on 16 vertices of the triangle-free Kneser graph $K(8,3)$ is not semi-transitive, and have raised the question on the existence of smaller triangle-free non-semi-transitive graphs. In this paper we prove that the smallest triangle-free 4-chromatic graph on 11 vertices (the Grotzsch graph) and the smallest triangle-free 4-chromatic 4-regular graph on 12 vertices (the Chvatal graph) are not semi-transitive. Hence, the Grotzsch graph is the smallest triangle-free non-semi-transitive graph. We also prove the existence of semi-transitive graphs of girth 4 with chromatic number 4 including a small one (the circulant graph $C(13;1,5)$ on 13 vertices) and dense ones (Tofts graphs). Finally, we show that each $4$-regular circulant graph (possibly containing triangles) is semi-transitive.
An adjacent vertex distinguishing coloring of a graph G is a proper edge coloring of G such that any pair of adjacent vertices are incident with distinct sets of colors. The minimum number of colors needed for an adjacent vertex distinguishing coloring of G is denoted by $chi_a(G)$. In this paper, we prove that $chi_a(G)$ <= 5($Delta+2$)/2 for any graph G having maximum degree $Delta$ and no isolated edges. This improves a result in [S. Akbari, H. Bidkhori, N. Nosrati, r-Strong edge colorings of graphs, Discrete Math. 306 (2006), 3005-3010], which states that $chi_a(G)$ <= 3$Delta$ for any graph G without isolated edges.
A tri-colored sum-free set in an abelian group $H$ is a collection of ordered triples in $H^3$, ${(a_i,b_i,c_i)}_{i=1}^m$, such that the equation $a_i+b_j+c_k=0$ holds if and only if $i=j=k$. Using a variant of the lemma introduced by Croot, Lev, and Pach in their breakthrough work on arithmetic-progression-free sets, we prove that the size of any tri-colored sum-free set in $mathbb{F}_2^n$ is bounded above by $6 {n choose lfloor n/3 rfloor}$. This upper bound is tight, up to a factor subexponential in $n$: there exist tri-colored sum-free sets in $mathbb{F}_2^n$ of size greater than ${n choose lfloor n/3 rfloor} cdot 2^{-sqrt{16 n / 3}}$ for all sufficiently large $n$.
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