Coloring intersection graphs of arc-connected sets in the plane

A family of sets in the plane is simple if the intersection of its any subfamily is arc-connected, and it is pierced by a line $L$ if the intersection of its any member with $L$ is a nonempty segment. It is proved that the intersection graphs of simple families of compact arc-connected sets in the plane pierced by a common line have chromatic number bounded by a function of their clique number.Comment: Minor changes + some additional references not included in the journal versio

Extremal Colorings and Independent Sets

We consider several extremal problems of maximizing the number of colorings and independent sets in some graph families with fixed chromatic number and order. First, we address the problem of maximizing the number of colorings in the family of connected graphs with chromatic number k and order n where k≥4 role= presentation style= box-sizing: inherit; display: inline; font-style: normal; font-weight: normal; line-height: normal; font-size: 18px; text-indent: 0px; text-align: left; text-transform: none; letter-spacing: normal; word-spacing: normal; overflow-wrap: normal; white-space: nowrap; float: none; direction: ltr; max-width: none; max-height: none; min-width: 0px; min-height: 0px; border: 0px; padding: 0px; margin: 0px; position: relative; \u3ek≥4k≥4. It was conjectured that extremal graphs are those which have clique number k and size (k2)+n−k role= presentation style= box-sizing: inherit; display: inline; font-style: normal; font-weight: normal; line-height: normal; font-size: 18px; text-indent: 0px; text-align: left; text-transform: none; letter-spacing: normal; word-spacing: normal; overflow-wrap: normal; white-space: nowrap; float: none; direction: ltr; max-width: none; max-height: none; min-width: 0px; min-height: 0px; border: 0px; padding: 0px; margin: 0px; position: relative; \u3e(k2)+n−k(k2)+n−k. We affirm this conjecture for 4-chromatic claw-free graphs and for all k-chromatic line graphs with k≥4 role= presentation style= box-sizing: inherit; display: inline; font-style: normal; font-weight: normal; line-height: normal; font-size: 18px; text-indent: 0px; text-align: left; text-transform: none; letter-spacing: normal; word-spacing: normal; overflow-wrap: normal; white-space: nowrap; float: none; direction: ltr; max-width: none; max-height: none; min-width: 0px; min-height: 0px; border: 0px; padding: 0px; margin: 0px; position: relative; \u3ek≥4k≥4. We also reduce this extremal problem to a finite family of graphs when restricted to claw-free graphs. Secondly, we determine the maximum number of independent sets of each size in the family of n-vertex k-chromatic graphs (respectively connected n-vertex k-chromatic graphs and n-vertex k-chromatic graphs with c components). We show that the unique extremal graph is Kk∪En−k role= presentation style= box-sizing: inherit; display: inline; font-style: normal; font-weight: normal; line-height: normal; font-size: 18px; text-indent: 0px; text-align: left; text-transform: none; letter-spacing: normal; word-spacing: normal; overflow-wrap: normal; white-space: nowrap; float: none; direction: ltr; max-width: none; max-height: none; min-width: 0px; min-height: 0px; border: 0px; padding: 0px; margin: 0px; position: relative; \u3eKk∪En−kKk∪En−k, K1∨(Kk−1∪En−k) role= presentation style= box-sizing: inherit; display: inline; font-style: normal; font-weight: normal; line-height: normal; font-size: 18px; text-indent: 0px; text-align: left; text-transform: none; letter-spacing: normal; word-spacing: normal; overflow-wrap: normal; white-space: nowrap; float: none; direction: ltr; max-width: none; max-height: none; min-width: 0px; min-height: 0px; border: 0px; padding: 0px; margin: 0px; position: relative; \u3eK1∨(Kk−1∪En−k)K1∨(Kk−1∪En−k) and (K1∨(Kk−1∪En−k−c+1))∪Ec−1 role= presentation style= box-sizing: inherit; display: inline; font-style: normal; font-weight: normal; line-height: normal; font-size: 18px; text-indent: 0px; text-align: left; text-transform: none; letter-spacing: normal; word-spacing: normal; overflow-wrap: normal; white-space: nowrap; float: none; direction: ltr; max-width: none; max-height: none; min-width: 0px; min-height: 0px; border: 0px; padding: 0px; margin: 0px; position: relative; \u3e(K1∨(Kk−1∪En−k−c+1))∪Ec−1(K1∨(Kk−1∪En−k−c+1))∪Ec−1 respectively

On colouring point visibility graphs

In this paper we show that it can be decided in polynomial time whether or not the visibility graph of a given point set is 4-colourable, and such a 4-colouring, if it exists, can also be constructed in polynomial time. We show that the problem of deciding whether the visibility graph of a point set is 5-colourable, is NP-complete. We give an example of a point visibility graph that has chromatic number 6 while its clique number is only 4

Perfect graphs of arbitrarily large clique-chromatic number

We prove that there exist perfect graphs of arbitrarily large clique-chromatic number. These graphs can be obtained from cobipartite graphs by repeatedly gluing along cliques. This negatively answers a question raised by Duffus, Sands, Sauer, and Woodrow in [Two-coloring all two-element maximal antichains, J. Combinatorial Theory, Ser. A, 57 (1991), 109-116]