Strong chromatic index of sparse graphs

A coloring of the edges of a graph $G$ is strong if each color class is an induced matching of $G$. The strong chromatic index of $G$, denoted by $\chi_{s}^{\prime}(G)$, is the least number of colors in a strong edge coloring of $G$. In this note we prove that $\chi_{s}^{\prime}(G)\leq (4k-1)\Delta (G)-k(2k+1)+1$ for every $k$-degenerate graph $G$. This confirms the strong version of conjecture stated recently by Chang and Narayanan [3]. Our approach allows also to improve the upper bound from [3] for chordless graphs. We get that $$ for any chordless graph $G$. Both bounds remain valid for the list version of the strong edge coloring of these graphs

3-Colourability of Dually Chordal Graphs in Linear Time

A graph G is dually chordal if there is a spanning tree T of G such that any maximal clique of G induces a subtree in T. This paper investigates the Colourability problem on dually chordal graphs. It will show that it is NP-complete in case of four colours and solvable in linear time with a simple algorithm in case of three colours. In addition, it will be shown that a dually chordal graph is 3-colourable if and only if it is perfect and has no clique of size four

Graphs that do not contain a cycle with a node that has at least two neighbors on it

We recall several known results about minimally 2-connected graphs, and show that they all follow from a decomposition theorem. Starting from an analogy with critically 2-connected graphs, we give structural characterizations of the classes of graphs that do not contain as a subgraph and as an induced subgraph, a cycle with a node that has at least two neighbors on the cycle. From these characterizations we get polynomial time recognition algorithms for these classes, as well as polynomial time algorithms for vertex-coloring and edge-coloring

On hereditary graph classes defined by forbidding Truemper configurations: recognition and combinatorial optimization algorithms, and χ-boundedness results

Truemper configurations are four types of graphs that helped us understand the structure of several well-known hereditary graph classes. The most famous examples are perhaps the class of perfect graphs and the class of even-hole-free graphs: for both of them, some Truemper configurations are excluded (as induced subgraphs), and this fact appeared to be useful, and played some role in the proof of the known decomposition theorems for these classes. The main goal of this thesis is to contribute to the systematic exploration of hereditary graph classes defined by forbidding Truemper configurations. We study many of these classes, and we investigate their structure by applying the decomposition method. We then use our structural results to analyze the complexity of the maximum clique, maximum stable set and optimal coloring problems restricted to these classes. Finally, we provide polynomial-time recognition algorithms for all of these classes, and we obtain χ-boundedness results

Strong chromatic index of k-degenerate graphs

A {\em strong edge coloring} of a graph $G$ is a proper edge coloring in which every color class is an induced matching. The {\em strong chromatic index} \chiup_{s}'(G) of a graph $G$ is the minimum number of colors in a strong edge coloring of $G$. In this note, we improve a result by D{\k e}bski \etal [Strong chromatic index of sparse graphs, arXiv:1301.1992v1] and show that the strong chromatic index of a $k$-degenerate graph $G$ is at most $(4k-2) \cdot \Delta(G) - 2k^{2} + 1$. As a direct consequence, the strong chromatic index of a $2$-degenerate graph $G$ is at most $6\Delta(G) - 7$, which improves the upper bound $10\Delta(G) - 10$ by Chang and Narayanan [Strong chromatic index of 2-degenerate graphs, J. Graph Theory 73 (2013) (2) 119--126]. For a special subclass of $2$-degenerate graphs, we obtain a better upper bound, namely if $G$ is a graph such that all of its $3^{+}$-vertices induce a forest, then \chiup_{s}'(G) \leq 4 \Delta(G) -3; as a corollary, every minimally $2$-connected graph $G$ has strong chromatic index at most $4 \Delta(G) - 3$. Moreover, all the results in this note are best possible in some sense.Comment: 3 pages in Discrete Mathematics, 201