Chromatic Number Of Triangle-free Graphs Research Articles

Coloring triangle-free graphs with local list sizes.

We prove two distinct and natural refinements of a recent breakthrough result of Molloy (and a follow‐up work of Bernshteyn) on the (list) chromatic number of triangle‐free graphs. In both our results, we permit the amount of color made available to vertices of lower degree to be accordingly lower. One result concerns list coloring and correspondence coloring, while the other concerns fractional coloring. Our proof of the second illustrates the use of the hard‐core model to prove a Johansson‐type result, which may be of independent interest.

Some Results on Chromatic Number as a Function of Triangle Count

A variety of powerful extremal results have been shown for the chromatic number of triangle-free graphs. Three noteworthy bounds are in terms of the number of vertices, edges, and maximum degree given by Poljak and Tuza [SIAM J. Discrete Math., 7 (1994), pp. 307--313] and Johansson. There have been comparatively fewer works extending these types of bounds to graphs with a small number of triangles. One noteworthy exception is a result of Alon, Krivelevich, and Sudakov [J. Combin. Theory Ser. B, 77 (1999), pp. 73--82] bounding the chromatic number for graphs with low degree and few triangles per vertex; this bound is nearly the same as for triangle-free graphs. This type of parametrization is much less rigid and has appeared in dozens of combinatorial constructions. In this paper, we show a similar type of result for $\chi(G)$ as a function of the number of vertices $n$, the number of edges $m$, as well as the triangle count (both local and global measures). Our results smoothly interpolate between the generic bounds true for all graphs and bounds for triangle-free graphs. Our results are tight for most of these cases; we show how an open problem regarding fractional chromatic number and degeneracy in triangle-free graphs can resolve the small remaining gap in our bounds.

Triangle-Free Subgraphs of Random Graphs

The Andrásfai–Erdős–Sós Theorem [Andrásfai, B., P. Erdős and V. T. Sós, On the connection between chromatic number, maximal clique and minimal degree of a graph, Discrete Math. 8 (1974), pp. 205–218] states that all triangle-free graphs on n vertices with minimum degree strictly greater than 2n/5 are bipartite. Thomassen [Thomassen, C., On the chromatic number of triangle-free graphs of large minimum degree, Combinatorica 22 (2002), pp. 591–596] proved that when the minimum degree condition is relaxed to (13+ε)n, the result is still guaranteed to be rε-partite, where rε does not depend on n. We prove best possible random graph analogues of these theorems.

Upper bounds on the chromatic number of triangle-free graphs with a forbidden subtree

Gyarfas conjectured that for a given forest F, there exists an integer function f(F, x) such that $$\chi (G)\le f(F,\omega (G))$$ź(G)≤f(F,ź(G)) for each F-free graph G, where $$\omega (G)$$ź(G) is the clique number of G. The broom B(m, n) is the tree of order $$m+n$$m+n obtained from identifying a vertex of degree 1 of the path $$P_m$$Pm with the center of the star $$K_{1,n}$$K1,n. In this note, we prove that every connected, triangle-free and B(m, n)-free graph is $$(m+n-2)$$(m+n-2)-colorable as an extension of a result of Randerath and Schiermeyer and a result of Gyarfas, Szemeredi and Tuza. In addition, it is also shown that every connected, triangle-free, $$C_4$$C4-free and T-free graph is $$(p-2)$$(p-2)-colorable, where T is a tree of order $$p\ge 4$$pź4 and $$T\not \cong K_{1,3}$$TźK1,3.

Distributed coloring algorithms for triangle-free graphs

Vertex coloring is a central concept in graph theory and an important symmetry-breaking primitive in distributed computing. Whereas degree-Δ graphs may require palettes of Δ+1 colors in the worst case, it is well known that the chromatic number of many natural graph classes can be much smaller. In this paper we give new distributed algorithms to find (Δ/k)-coloring in graphs of girth 4 (triangle-free graphs), girth 5, and trees. The parameter k can be at most (14−o(1))ln⁡Δ in triangle-free graphs and at most (1−o(1))ln⁡Δ in girth-5 graphs and trees, where o(1) is a function of Δ. Specifically, for Δ sufficiently large we can find such a coloring in O(k+log⁎⁡n) time. Moreover, for any Δ we can compute such colorings in roughly logarithmic time for triangle-free and girth-5 graphs, and in O(log⁡Δ+logΔ⁡log⁡n) time on trees. As a byproduct, our algorithm shows that the chromatic number of triangle-free graphs is at most (4+o(1))Δln⁡Δ, which improves on Jamall's recent bound of (67+o(1))Δln⁡Δ. Finally, we show that (Δ+1)-coloring for triangle-free graphs can be obtained in sublogarithmic time for any Δ.

The fractional chromatic number of triangle-free graphs with [formula omitted

Let G be a triangle-free graph with maximum degree at most 3. Staton proved that the independence number of G is at least 514|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)≤145. Recently, Hatami and Zhu proved that χf(G)≤3−364. In this paper, we prove χf(G)≤3−343.

On the Chromatic Number of Triangle-Free Graphs of Large Minimum Degree

We prove that, for each fixed real number c > 1/3, the triangle-free graphs of minimum degree at least cn (where n is the number of vertices) have bounded chromatic number. This problem was raised by Erdős and Simonovits in 1973 who pointed out that there is no such result for c < 1/3.

Triangle-free graphs and forbidden subgraphs

The relation of chromatic aspects and the existence of certain induced subgraphs of a triangle-free graph will be investigated. Based on a characterization statement of Pach, some results on the chromatic number of triangle-free graphs with certain forbidden induced subgraphs will be refined by describing their structure in terms of homomorphisms. In particular, we introduce chordal triangle-free graphs as a natural superclass of chordal bipartite graphs and describe the structure of the maximal triangle-free members. Finally, we improve on the upper bound for the chromatic number of triangle-free sK 2-free graphs by 1 for s⩾2 giving the tight bound for s=3.