We give an infinite class of counterexamples to the Gotsman-Linial conjecture when d = 2. On the other hand, we establish an asymptotic form of the conjecture for quadratic threshold functions whose non-zero quadratic terms define a graph with either low fractional chromatic number or few edges. Our techniques are elementary and our exposition is self-contained, if you're into that.

Reed conjectured that for every ε > 0 and every integer ∆, there exists g such that the fractional total chromatic number of every graph with maximum degree ∆ and girth at least g is at most ∆ + 1 + ε. The conjecture was proven to be true when ∆ = 3 or ∆ is even. We settle the conjecture by proving it for the remaining cases.

The concepts of (k, d)-coloring and the star chromatic number, studied by Vince, by Bondy and Hell, and by Zhu are shown to reflect the cographic instance of a wider concept, that of fractional nowhere-zero flows in regular matroids. c © 1998 John Wiley & Sons, Inc. J Graph Theory 28: 155–161, 1998

The development of graph theory has provided many new pieces knowledge, one them is color. Where the application spread in various fields such as coding index theory. Fractional coloring multiple at points with different colors where adjoining point a operation known sum operation. Point can be applied to graphs result operations from several special graphs. In this case, summation results path...

For loopless multigraphs, the total chromatic number is asymptotically its fractional counterpart as the latter invariant tends to infinity. The proof of this is based on a recent theorem of Kahn establishing the analogous asymptotic behaviour of the list-chromatic index for multigraphs. The total colouring conjecture, proposed independently by Behzad [1] and Vizing [11], asserts that the total...

The packing chromatic number χρ(G) of a graphG is the smallest integer k such that the vertex set of G can be partitioned into sets Vi, i ∈ [k], where each Vi is an i-packing. In this paper, we investigate for a given triple (a, b, c) of positive integers whether there exists a graph G such that ω(G) = a, χ(G) = b, and χρ(G) = c. If so, we say that (a, b, c) is realizable. It is proved that b =...

Let G be a triangle-free graph with maximum degree at most 3. Staton proved that the independence number of G is at least 5 14 |V (G)|. Heckman and Thomas conjectured that Staton’s result can be strengthened into a bound on the fractional chromatic number of G, namely χf (G) ≤ 14 5 . Recently, Hatami and Zhu proved that χf (G) ≤ 3 − 3 64 . In this paper, we prove χf (G) ≤ 3 − 3 43 . © 2012 Else...

We present a necessary and sufficient condition for a graph of odd-girth 2k + 1 to bound the class of K4-minor-free graphs of odd-girth (at least) 2k + 1, that is, to admit a homomorphism from any such K4-minor-free graph. This yields a polynomial-time algorithm to recognize such bounds. Using this condition, we first prove that every K4-minor free graph of odd-girth 2k+1 admits a homomorphism ...